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 Solutions to Problems 
P4-1. LG 1: Using a time line 
Basic 
a, b, and c 
d. Financial managers rely more on present value than future value because they typically make 
decisions before the start of a project, at time zero, as does the present value calculation. 
P4-2. LG 2: Future value calculation: FVn = PV ´ (1 + I)n 
Basic 
Case 
A FVIF12%,2 periods = (1 + 0.12)2 = 1.254 
B FVIF6%,3 periods= (1 + 0.06)3 = 1.191 
C FVIF9%,2 periods= (1 + 0.09)2 = 1.188 
D FVIF3%,4 periods= (1 + 0.03)4 = 1.126 
P4-3. LG 2: Future value tables: FVn = PV ´ (1 + I)n 
Basic 
Case A 
a. 2 = 1 ´ (1 + 0.07)n b. 4 = 1 ´ (1 + 0.07)n 
2/1 = (1.07)n 4/1 = (1.07)n 
2 = FVIF7%,n 4 = FVIF7%,n 
10 years < n < 11 years 20 years < n < 21 years 
Nearest to 10 years Nearest to 20 years 
Case B
73 Gitman • Principles of Managerial Finance, Twelfth Edition 
a. 2 = 1 ´ (1 + 0.40)n b. 4 = (1 + 0.40)n 
2 = FVIF40%,n 4 = FVIF40%,n 
2 years < n < 3 years 4 years < n < 5 years 
Nearest to 2 years Nearest to 4 years 
Case C 
a. 2 = 1 ´ (1 + 0.20)n b. 4 = (1 + 0.20)n 
2 = FVIF20%,n 4 = FVIF20%,n 
3 years < n < 4 years 7 years < n < 8 years 
Nearest to 4 years Nearest to 8 years 
Case D 
a. 2 = 1 ´ (1 + 0.10)n b. 4 = (1 + 0.10)n 
2 = FVIF10%,n 4 = FVIF40%,n 
7 years < n < 8 years 14 years < n <15 years 
Nearest to 7 years Nearest to 15 years 
P4-4. LG 2: Future values: FVn = PV ´ (1 + I)n or FVn = PV ´ (FVIFi%,n) 
Intermediate 
Case Case 
A FV20 = PV ´ FVIF5%,20 yrs. B FV7 = PV ´ FVIF8%,7 yrs. 
FV20 = $200 ´ (2.653) FV7 = $4,500 ´ (1.714) 
FV20 = $530.60 FV7 = $7,713 
Calculator solution: $530.66 Calculator solution: $7,712.21
Chapter 4 Time Value of Money 74 
C FV10 = PV ´ FVIF9%,10 yrs. D FV12 = PV ´ FVIF10%,12 yrs. 
FV10 = $10,000 ´ (2.367) FV12 = $25,000 ´ (3.138) 
FV10 = $23,670 FV12 =  $78,450 
Calculator solution: $23,673.64 Calculator solution: $78,460.71 
E FV5 = PV ´ FVIF11%,5 yrs. F FV9 = PV ´ FVIF12%,9 yrs. 
FV5 = $37,000 ´ (1.685) FV9 = $40,000 ´ (2.773) 
FV5 = $62,345 FV9 = $110,920 
Calculator solution: $62,347.15 Calculator solution: $110,923.15 
P4-5. LG 2: Personal finance: Time value: FVn = PV ´ (1 + I)n or FVn = PV ´ (FVIFi%,n) 
Intermediate 
a. (1) FV3 = PV ´ (FVIF7%,3) b. (1) Interest earned = FV3 – PV 
FV3 = $1,500 ´ (1.225) Interest earned = $1,837.50 
FV3 = $1,837.50 –$1,500.00 
Calculator solution: $1,837.56 $337.50 
(2) FV6 = PV ´ (FVIF7%,6) (2) Interest earned = FV6 – FV3 
FV6 = $1,500 ´ (1.501) Interest earned = $2,251.50 
FV6 = $2,251.50 –$1,837.50 
Calculator solution: $2,251.10 $414.00 
(3) FV9 = PV ´ (FVIF7%,9) (3) Interest earned = FV9 – FV6 
FV9 = $1,500 ´ (1.838) Interest earned = $2,757.00 
FV9 = $2,757.00 –$2,251.50 
Calculator solution: $2,757.69 $505.50 
c. The fact that the longer the investment period is, the larger the total amount of interest 
collected will be, is not unexpected and is due to the greater length of time that the principal 
sum of $1,500 is invested. The most significant point is that the incremental interest earned 
per 3-year period increases with each subsequent 3 year period. The total interest for the first 
3 years is $337.50; however, for the second 3 years (from year 3 to 6) the additional interest 
earned is $414.00. For the third 3-year period, the incremental interest is $505.50. This 
increasing change in interest earned is due to compounding, the earning of interest on 
previous interest earned. The greater the previous interest earned, the greater the impact of 
compounding. 
P4-6. LG 2: Personal finance: Time value 
Challenge 
a. (1) FV5 = PV ´ (FVIF2%,5) (2) FV5 = PV ´ (FVIF4%,5) 
FV5 = $14,000 ´ (1.104) FV5 = $14,000 ´ (1.217)
75 Gitman • Principles of Managerial Finance, Twelfth Edition 
FV5 = $15,456.00 FV5 = $17,038.00 
Calculator solution: $15,457.13 Calculator solution: $17,033.14 
b. The car will cost $1,582 more with a 4% inflation rate than an inflation rate of 2%. This 
increase is 10.2% more ($1,582 ¸ $15,456) than would be paid with only a 2% rate of 
inflation. 
P4-7. LG 2: Personal finance: Time value 
Challenge 
Deposit Now: Deposit in 10 Years: 
FV40 = PV ´ FVIF9%,40 FV30 = PV10 ´ (FVIF9%,30) 
FV40 = $10,000 ´ (1.09)40 FV30 = PV10 ´ (1.09)30 
FV40 = $10,000 ´ (31.409) FV30 = $10,000 ´ (13.268) 
FV40 = $314,090.00 FV30 = $132,680.00 
Calculator solution: $314,094.20 Calculator solution: $132,676.78 
You would be better off by $181,410 ($314,090 – $132,680) by investing the $10,000 now 
instead of waiting for 10 years to make the investment. 
P4-8. LG 2: Personal finance: Time value: FVn = PV ´ FVIFi%,n 
Challenge 
a. $15,000 = $10,200 ´ FVIFi%,5 b. $15,000 = $8,150 ´ FVIFi%,5 
FVIFi%,5 = $15,000 ¸ $10,200 = 1.471 FVIFi%,5 = $15,000 ¸ $8,150 = 1.840 
8% < i < 9% 12% < i < 13% 
Calculator solution: 8.02% Calculator solution: 12.98% 
c. $15,000 = $7,150 ´ FVIFi%,5 
FVIFi%,5 = $15,000 ¸ $7,150 = 2.098 
15% < i < 16% 
Calculator solution: 15.97% 
P4-9. LG 2: Personal finance: Single-payment loan repayment: FVn = PV ´ FVIFi%,n 
Intermediate 
a. FV1 = PV ´ (FVIF14%,1) b. FV4 = PV ´ (FVIF14%,4) 
FV1 = $200 ´ (1.14) FV4 = $200 ´ (1.689) 
FV1 = $228 FV4 = $337.80
Chapter 4 Time Value of Money 76 
Calculator solution: $228 Calculator solution: $337.79 
c. FV8 = PV ´ (FVIF14%,8) 
FV8 = $200 ´ (2.853) 
FV8 = $570.60 
Calculator solution: $570.52 
P4-10. LG 2: Present value calculation: = 
1 
+ 
PVIF 
(1 i)n 
Basic 
Case 
A PVIF = 1 ¸ (1 + 0.02)4 = 0.9238 
B PVIF = 1 ¸ (1 + 0.10)2 = 0.8264 
C PVIF = 1 ¸ (1 + 0.05)3 = 0.8638 
D PVIF = 1 ¸ (1 + 0.13)2 = 0.7831 
P4-11. LG 2: Present values: PV = FVn ´ (PVIFi%,n) 
Basic 
Case Calculator Solution 
A PV12%,4yrs = $7,000 ´ 0.636 = $4,452 $4,448.63 
B PV8%, 20yrs = $28,000 ´ 0.215 = $6,020 $6,007.35 
C PV14%,12yrs = $10,000 ´ 0.208 = $2,080 $2,075.59 
D PV11%,6yrs = $150,000 ´ 0.535 = $80,250 $80,196.13 
E PV20%,8yrs = $45,000 ´ 0.233 = $10,485 $10,465.56
77 Gitman • Principles of Managerial Finance, Twelfth Edition 
P4-12. LG 2: Present value concept: PVn = FVn ´ (PVIFi%,n) 
Intermediate 
a. PV = FV6 ´ (PVIF12%,6) b. PV = FV6 ´ (PVIF12%,6) 
PV = $6,000 ´ (.507) PV = $6,000 ´ (0.507) 
PV = $3,042.00 PV = $3,042.00 
Calculator solution: $3,039.79 Calculator solution: $3,039.79 
c. PV = FV6 ´ (PVIF12%,6) 
PV = $6,000 ´ (0.507) 
PV = $3,042.00 
Calculator solution: $3,039.79 
d. The answer to all three parts are the same. In each case the same questions is being asked but 
in a different way. 
P4-13. LG 2: Personal finance: Time value: PV = FVn ´ (PVIFi%,n) 
Basic 
Jim should be willing to pay no more than $408.00 for this future sum given that his opportunity 
cost is 7%. 
PV = $500 ´ (PVIF7%,3) 
PV = $500 ´ (0.816) 
PV = $408.00 
Calculator solution: $408.15 
P4-14. LG 2: Time value: PV = FVn ´ (PVIFi%,n) 
Intermediate 
PV = $100 ´ (PVIF8%,6) 
PV = $100 ´ (0.630) 
PV = $63.00 
Calculator solution: $63.02
Chapter 4 Time Value of Money 78 
P4-15. LG 2: Personal finance: Time value and discount rates: PV = FVn ´ (PVIFi%,n) 
Intermediate 
a. (1) PV = $1,000,000 ´ (PVIF6%,10) (2) PV = $1,000,000 ´ (PVIF9%,10) 
PV = $1,000,000 ´ (0.558) PV = $1,000,000 ´ (0.422) 
PV = $558,000.00 PV = $422,000.00 
Calculator solution: $558,394.78 Calculator solution: $422,410.81 
(3) PV = $1,000,000 ´ (PVIF12%,10) 
PV = $1,000,000 ´ (0.322) 
PV = $322,000.00 
Calculator solution: $321,973.24 
b. (1) PV = $1,000,000 ´ (PVIF6%,15) (2) PV = $1,000,000 ´ (PVIF9%,15) 
PV = $1,000,000 ´ (0.417) PV = $1,000,000 ´ (0.275) 
PV = $417,000.00 PV = $275,000.00 
Calculator solution: $417,265.06 Calculator solution: $274,538.04 
(3) PV = $1,000,000 ´ (PVIF12%,15) 
PV = $1,000,000 ´ (0.183) 
PV = $183,000.00 
Calculator solution: $182,696.26 
c. As the discount rate increases, the present value becomes smaller. This decrease is due to 
the higher opportunity cost associated with the higher rate. Also, the longer the time until the 
lottery payment is collected, the less the present value due to the greater time over which the 
opportunity cost applies. In other words, the larger the discount rate and the longer the time 
until the money is received, the smaller will be the present value of a future payment. 
P4-16. Personal finance: LG 2: Time value comparisons of lump sums: PV = FVn ´ (PVIFi%,n) 
Intermediate 
a. A PV = $28,500 ´ (PVIF11%,3) B PV = $54,000 ´ (PVIF11%,9) 
PV = $28,500 ´ (0.731) PV = $54,000 ´ (0.391) 
PV = $20,833.50 PV = $21,114.00 
Calculator solution: $20,838.95 Calculator solution: $21,109.94 
C PV = $160,000 ´ (PVIF11%,20) 
PV = $160,000 ´ (0.124) 
PV = $19,840.00 
Calculator solution: $19,845.43 
b. Alternatives A and B are both worth greater than $20,000 in term of the present value. 
c. The best alternative is B because the present value of B is larger than either A or C and is also 
greater than the $20,000 offer.
79 Gitman • Principles of Managerial Finance, Twelfth Edition 
P4-17. LG 2: Personal finance: Cash flow investment decision: PV = FVn ´ (PVIFi%,n) 
Intermediate 
A PV = $30,000 ´ (PVIF10%,5) B PV = $3,000 ´ (PVIF10%,20) 
PV = $30,000 ´ (0.621) PV = $3,000 ´ (0.149) 
PV = $18,630.00 PV = $447.00 
Calculator solution: $18,627.64 Calculator solution: $445.93 
C PV = $10,000 ´ (PVIF10%,10) D PV = $15,000 ´ (PVIF10%,40) 
PV = $10,000 ´ (0.386) PV = $15,000 ´ (0.022) 
PV = $3,860.00 PV = $330.00 
Calculator solution: $3,855.43 Calculator solution: $331.42 
Purchase Do Not Purchase 
A B 
C D 
P4-18. LG 3: Future value of an annuity 
Intermediate 
a. Future value of an ordinary annuity vs. annuity due 
(1) Ordinary Annuity (2) Annuity Due 
FVAk%,n = PMT ´ (FVIFAk%,n) FVAdue = PMT ´ [(FVIFAk%,n ´ (1 + k)] 
A FVA8%,10 = $2,500 ´ 14.487 FVAdue = $2,500 ´ (14.487 ´ 1.08) 
FVA8%,10 = $36,217.50 FVAdue = $39,114.90 
Calculator solution: $36,216.41 Calculator solution: $39,113.72 
B FVA12%,6 = $500 ´ 8.115 FVAdue = $500 ´( 8.115 ´ 1.12) 
FVA12%,6 = $4,057.50 FVAdue = $4,544.40 
Calculator solution: $4,057.59 Calculator solution: $4,544.51 
C FVA20%,5 = $30,000 ´ 7.442 FVAdue = $30,000 ´ (7.442 ´ 1.20) 
FVA20%,5 = $223,260 FVAdue = $267,912 
Calculator solution: $223,248 Calculator solution: $267,897.60 
D FVA9%,8 = $11,500 ´ 11.028 FVAdue = $11,500 ´ (11.028 ´ 1.09) 
FVA9%,8 = $126,822 FVAdue = $138,235.98 
Calculator solution: $126,827.45 Calculator solution: $138,241.92 
E FVA14%,30 = $6,000 ´ 356.787 FVAdue = $6,000 ´ (356.787 ´ 1.14) 
FVA14%,30 = $2,140,722 FVAdue = $2,440,422.00 
Calculator solution: $2,140,721.08 Calculator solution: $2,440,422.03 
b. The annuity due results in a greater future value in each case. By depositing the payment at 
the beginning rather than at the end of the year, it has one additional year of compounding.
Chapter 4 Time Value of Money 80 
P4-19. LG 3: Present value of an annuity: PVn = PMT ´ (PVIFAi%,n) 
Intermediate 
a. Present value of an ordinary annuity vs. annuity due 
(1) Ordinary Annuity (2) Annuity Due 
PVAk%,n = PMT ´ (PVIFAi%,n) PVAdue = PMT ´ [(PVIFAi%,n ´ (1 + k)] 
A PVA7%,3 = $12,000 ´ 2.624 PVAdue = $12,000 ´ (2.624 ´ 1.07) 
PVA7%,3 = $31,488 PVAdue = $33,692 
Calculator solution: $31,491.79 Calculator solution: $33,696.22 
B PVA12%15 = $55,000 ´ 6.811 PVAdue = $55,000 ´ (6.811 ´ 1.12) 
PVA12%,15 = $374,605 PVAdue = $419,557.60 
Calculator solution: $374,597.55 Calculator solution: $419,549.25 
C PVA20%,9 = $700 ´ 4.031 PVAdue = $700 ´ (4.031 ´ 1.20) 
PVA20%,9 = $2,821.70 PVAdue = $3,386.04 
Calculator solution: $2,821.68 Calculator solution: $3,386.01 
D PVA5%,7 = $140,000 ´ 5.786 PVAdue = $140,000 ´ (5.786 ´ 1.05) 
PVA5%,7 = $810,040 PVAdue = $850,542 
Calculator solution: $810,092.28 Calculator solution: $850,596.89 
E PVA10%,5 = $22,500 ´ 3.791 PVAdue = $22,500 ´ (2.791 ´ 1.10) 
PVA10%,5 = $85,297.50 PVAdue = $93,827.25 
Calculator solution: $85,292.70 Calculator solution: $93,821.97 
b. The annuity due results in a greater present value in each case. By depositing the payment at 
the beginning rather than at the end of the year, it has one less year to discount back. 
P4-20. LG 3: Personal finance: Time value–annuities 
Challenge 
a. Annuity C (Ordinary) Annuity D (Due) 
FVAi%,n = PMT ´ (FVIFAi%,n) FVAdue = PMT ´ [FVIFAi%,n ´ (1 + i)] 
(1) FVA10%,10 = $2,500 ´ 15.937 FVAdue = $2,200 ´ (15.937 ´ 1.10) 
FVA10%,10 = $39,842.50 FVAdue = $38,567.54 
Calculator solution: $39,843.56 Calculator solution: $38,568.57 
(2) FVA20%,10 = $2,500 ´ 25.959 FVAdue = $2,200 ´ (25.959 ´ 1.20) 
FVA20%,10 = $64,897.50 FVAdue = $68,531.76 
Calculator solution: $64,896.71 Calculator solution: $68,530.92
81 Gitman • Principles of Managerial Finance, Twelfth Edition 
b. (1) At the end of year 10, at a rate of 10%, Annuity C has a greater value ($39,842.50 vs. 
$38,567.54). 
(2) At the end of year 10, at a rate of 20%, Annuity D has a greater value ($68,531.76 vs. 
$64,897.50). 
c. Annuity C (Ordinary) Annuity D (Due) 
PVAi%,n = PMT ´ (FVIFAi%,n) PVAdue = PMT ´ [FVIFAi%,n ´ (1 + i)] 
(1) PVA10%,10 = $2,500 ´ 6.145 PVAdue = $2,200 ´ (6.145 ´ 1.10) 
PVA10%,10 = $15,362.50 PVAdue = $14,870.90 
Calculator solution: $15,361.42 Calculator solution: $14,869.85 
(2) PVA20%,10 = $2,500 ´ 4.192 PVAdue = $2,200 ´ (4.192 ´ 1.20) 
PVA20%,10 = $10,480 PVAdue = $11,066.88 
Calculator solution: $10,481.18 Calculator solution: $11,068.13 
d. (1) At the beginning of the 10 years, at a rate of 10%, Annuity C has a greater value 
($15,362.50 vs. $14,870.90). 
(2) At the beginning of the 10 years, at a rate of 20%, Annuity D has a greater value 
($11,066.88 vs. $10,480.00). 
e. Annuity C, with an annual payment of $2,500 made at the end of the year, has a higher present 
value at 10% than Annuity D with an annual payment of $2,200 made at the beginning of the 
year. When the rate is increased to 20%, the shorter period of time to discount at the higher 
rate results in a larger value for Annuity D, despite the lower payment. 
P4-21. LG 3: Personal finance: Retirement planning 
Challenge 
a. FVA40 = $2,000 ´ (FVIFA10%,40) b. FVA30 = $2,000 ´ (FVIFA10%,30) 
FVA40 = $2,000 ´ (442.593) FVA30 = $2,000 ´ (164.494) 
FVA40 = $885,186 FVA30 = $328,988 
Calculator solution: $885,185.11 Calculator solution: $328,988.05 
c. By delaying the deposits by 10 years the total opportunity cost is $556,198. This difference is 
due to both the lost deposits of $20,000 ($2,000 ´ 10yrs.) and the lost compounding of 
interest on all of the money for 10 years. 
d. Annuity Due: 
FVA40 = $2,000 ´ (FVIFA10%,40) ´ (1 + 0.10) 
FVA40 = $2,000 ´ (486.852) 
FVA40 = $973,704 
Calculator solution: $973,703.62
Chapter 4 Time Value of Money 82 
FVA30 = $2,000 ´ (FVIFA10%,30) ´ (1.10) 
FVA30 = $2,000 ´ (180.943) 
FVA30 = $361,886 
Calculator solution: $361,886.85 
Both deposits increased due to the extra year of compounding from the beginning-of-year 
deposits instead of the end-of-year deposits. However, the incremental change in the 40 year 
annuity is much larger than the incremental compounding on the 30 year deposit ($88,518 
versus $32,898) due to the larger sum on which the last year of compounding occurs. 
P4-22. LG 3: Personal finance: Value of a retirement annuity 
Intermediate 
PVA = PMT ´ (PVIFA9%,25) 
PVA = $12,000 ´ (9.823) 
PVA = $117,876.00 
Calculator solution: $117,870.96 
P4-23. LG 3: Personal finance: Funding your retirement 
Challenge 
a. PVA = PMT ´ (PVIFA11%,30) b. PV = FV ´ (PVIF9%,20) 
PVA = $20,000 ´ (8.694) PV = $173,880 ´ (0.178) 
PVA = $173,880.00 PV = $30,950.64 
Calculator solution: $173,875.85 Calculator solution: $31,024.82 
c. Both values would be lower. In other words, a smaller sum would be needed in 20 years for 
the annuity and a smaller amount would have to be put away today to accumulate the needed 
future sum. 
P4-24. LG 2, 3: Personal finance: Value of an annuity versus a single amount 
Intermediate 
a. PVAn = PMT ´ (PVIFAi%,n) 
PVA25 = $40,000 ´ (PVIFA5%,25) 
PVA25 = $40,000 ´ 14.094 
PVA25 = $563,760 
Calculator solution: $563,757.78 
At 5%, taking the award as an annuity is better; the present value is $563,760, compared to 
receiving $500,000 as a lump sum.
83 Gitman • Principles of Managerial Finance, Twelfth Edition 
b. PVAn = $40,000 ´ (PVIFA7%,25) 
PVA25 = $40,000 ´ (11.654) 
PVA25 = $466,160 
Calculator solution: $466,143.33 
At 7%, taking the award as a lump sum is better; the present value of the annuity is only 
$466,160, compared to the $500,000 lump sum payment. 
c. Because the annuity is worth more than the lump sum at 5% and less at 7%, try 6%: 
PV25 = $40,000 ´ (PVIFA6%,25) 
PV25 = $40,000 ´ 12.783 
PV25 = $511,320 
The rate at which you would be indifferent is greater than 6%; about 6.25% Calculator 
solution: 6.24% 
P4-25. LG 3: Perpetuities: PVn = PMT ´ (PVIFAi%,¥) 
Basic 
a. b. 
Case PV Factor PMT ´  (PVIFAi%,¥) = PMT ´  (1 ¸ 
i) 
A 1 ¸ 0.08 = 12.50 $20,000 ´ 12.50 = $250,000 
B 1 ¸ 0.10 = 10.00 $100,000 ´ 10.00 = $1,000,000 
C 1 ¸ 0.06 = 16.67 $3,000 ´ 16.67 = $50,000 
D 1 ¸ 0.05 = 20.00 $60,000 ´ 20.00 = $1,200,000 
P4-26. LG 3: Personal finance: Creating an endowment 
Intermediate 
a. PV = PMT ´ (PVIFAi%,¥) b. PV = PMT ´ (PVIFAi%,¥) 
PV = ($600 ´ 3) ´ (1 ¸ i) PV = ($600 ´ 3) ´ (1 ¸ i) 
PV = $1,800 ´ (1 ¸ 0.06) PV = $1,800 ´ (1 ¸ 0.09) 
PV = $1,800 ´ (16.67) PV = $1,800 ´ (11.11) 
PV = $30,006 PV = $19,998 
Calculator solution: $30,000 Calculator solution: $20,000
Chapter 4 Time Value of Money 84 
P4-27. LG 4: Value of a mixed stream 
Challenge 
a. 
Cash Flow 
Stream Year 
Number of Years 
to Compound FV = CF ´  FVIF12%,n Future Value 
A 1 3 $ 900 ´ 1.405 =  $ 1,264.50 
2 2 1,000 ´ 1.254 = 1,254.00 
3 1 1,200 ´ 1.120 = 1,344 .00 
$ 3,862 .50 
Calculator solution: $ 3,862.84 
B 1 5 $30,000 ´ 1.762 = $ 52,860.00 
2 4 25,000 ´ 1.574 = 39,350.00 
3 3 20,000 ´ 1.405 = 28,100.00 
4 2 10,000 ´ 1.254 = 12,540.00 
5 1 5,000 ´ 1.120 = 5,600 .00 
$138,450 .00 
Calculator solution: $138,450.79. 
C 1 4 $ 1,200 ´ 1.574 = $ 1,888.80 
2 3 1,200 ´ 1.405 = 1,686.00 
3 2 1,000 ´ 1.254 = 1,254.00 
4 1 1,900 ´ 1.120 = 2,128 .00 
$ 6,956 .80 
Calculator solution: $ 6,956.54 
b. If payments are made at the beginning of each period the present value of each of the end-of-period 
cash flow streams will be multiplied by (1 + i) to get the present value of the beginning-of- 
period cash flows. 
A $3,862.50 (1 + 0.12) = $4,326.00 
B $138,450.00 (1 + 0.12) = $155,064.00 
C $6,956.80 (1 + 0.12) = $7,791.62 
P4-28. LG 4: Personal finance: Value of a single amount versus a mixed stream 
Intermediate 
Lump Sum Deposit 
FV5 = PV ´ (FVIF7%,5)) 
FV5 = $24,000 ´ (1.403) 
FV5 = $33,672.00 
Calculator solution: $33,661.24
85 Gitman • Principles of Managerial Finance, Twelfth Edition 
Mixed Stream of Payments 
Beginning of 
Year 
Number of Years 
to Compound FV = CF ´  FVIF7%,n Future Value 
1 5 $ 2,000 ´ 1.403 = $ 2,805.00 
2 4 $ 4,000 ´ 1.311 = $ 5,243.00 
3 3 $ 6,000 ´ 1.225 = $ 7,350.00 
4 2 $ 8,000 ´ 1.145 = $ 9,159.00 
5 1 $10,000 ´ 1.070 = $10,700 .00 
$35,257 .00 
Calculator solution: $35,257.75 
Gina should select the stream of payments over the front-end lump sum payment. Her future 
wealth will be higher by $1,588. 
P4-29. LG 4: Value of mixed stream 
Basic 
Cash Flow 
Stream Year CF ´ PVIF12%,n = Present Value 
A 1 –$2000 ´ 0.893 = – $1,786 
2 3,000 ´ 0.797 = 2,391 
3 4,000 ´ 0.712 = 2,848 
4 6,000 ´ 0.636 = 3,816 
5 8,000 ´ 0.567 = 4,536 
$11,805 
Calculator solution: $11,805.51 
B 1 $10,000 ´ 0.893 = $ 8,930 
2–5 5,000 ´ 2.712a = 13,560 
6 7,000 ´ 0.507 = 3,549 
$26,039 
Calculator solution: $26,034.58 
*Sum of PV factors for years 2–5 
C 1–5 $10,000 ´ 3.605b $36,050 
6–10 8,000 ´ 2.045c 16,360 
$52,410 
Calculator solution: $52,411.34 
a PVIFA for 12% over years 2 through 5 = (PVIFA 12% 5 years) – (PVIFA 12% 1 year) 
b PVIFA for 12% 5 years 
c (PVIFA for 12%,10 years) – (PVIFA for 12%,5 years)
Chapter 4 Time Value of Money 86 
P4-30. LG 4: PV-mixed stream 
Intermediate 
a. 
Cash Flow 
Stream Year CF ´ PVIF15%,n = Present Value 
A 1 $50,000 ´ 0.870 = $43,500 
2 40,000 ´ 0.756 = 30,240 
3 30,000 ´ 0.658 = 19,740 
4 20,000 ´ 0.572 = 11,440 
5 10,000 ´ 0.497 = 4,970 
$109,890 
Calculator solution: $109,856.33 
B 1 $10,000 ´ 0.870 = $ 8,700 
2 20,000 ´ 0.756 = 15,120 
3 30,000 ´ 0.658 = 19,740 
4 40,000 ´ 0.572 = 22,880 
5 50,000 ´ 0.497 = 24,850 
$ 91,290 
Calculator solution: $ 91,272.98 
b. Cash flow stream A, with a present value of $109,890, is higher than cash flow stream B’s 
present value of $91,290 because the larger cash inflows occur in A in the early years when 
their present value is greater, while the smaller cash flows are received further in the future. 
P4-31. LG 1, 4: Value of a mixed stream 
Intermediate 
a.
87 Gitman • Principles of Managerial Finance, Twelfth Edition 
b. 
Cash Flow 
Stream Year CF ´ PVIF12%,n = Present Value 
A 1 $30,000 ´ 0.893 = $ 26,790 
2 25,000 ´ 0.797 = 19,925 
3–9 15,000 ´ 3.639* = 54,585 
10 10,000 ´ 0.322 = 3,220 
$104,520 
Calculator solution: $104,508.28 
* The PVIF for this 7-year annuity is obtained by summing together the PVIFs of 12% for periods 3 
through 9. This factor can also be calculated by taking the PVIFA12%,7 and multiplying by the 
PVIF12%,2. Alternatively, one could subtract PVIFA12%,2 from PVIFA12%,9. 
c. Harte should accept the series of payments offer. The present value of that mixed stream of 
payments is greater than the $100,000 immediate payment. 
P4-32. LG 5: Personal finance: Funding budget shortfalls 
Intermediate 
a. 
Year 
Budget 
Shortfall ´ PVIF8%,n = Present Value 
1 $5,000 ´ 0.926 = $ 4,630 
2 4,000 ´ 0.857 = 3,428 
3 6,000 ´ 0.794 = 4,764 
4 10,000 ´ 0.735 = 7,350 
5 3,000 ´ 0.681 = 2,043 
$22,215 
Calculator solution: $22,214.03 
A deposit of $22,215 would be needed to fund the shortfall for the pattern shown in the table. 
b. An increase in the earnings rate would reduce the amount calculated in part (a). The higher 
rate would lead to a larger interest being earned each year on the investment. The larger interest 
amounts will permit a decrease in the initial investment to obtain the same future value 
available for covering the shortfall.
Chapter 4 Time Value of Money 88 
P4-33. LG 4: Relationship between future value and present value-mixed stream 
Intermediate 
a. Present Value 
Year CF ´ PVIF5%,n = Present Value 
1 $800 ´ 0.952 = $ 761.60 
2 900 ´ 0.907 = 816.30 
3 1,000 ´ 0.864 = 864.00 
4 1,500 ´ 0.822 = 1,233.00 
5 2,000 ´ 0.784 = 1,568.00 
$5,242.90 
Calculator solution: $5,243.17 
b. The maximum you should pay is $5,242.90. 
c. A higher 7% discount rate will cause the present value of the cash flow stream to be lower 
than $5,242.90. 
P4-34. LG 5: Changing compounding frequency 
Intermediate 
a. Compounding frequency: FVn = PV ´ FVIFi%/m,n × m 
(1) Annual Semiannual 
12%, 5 years 12% ¸ 2 = 6%, 5 ´ 2 = 10 periods 
FV5 = $5,000 ´ (1.762) FV5 = $5,000 ´ (1.791) 
FV5 = $8,810 FV5 = $8,955 
Calculator solution: $8,811.71 Calculator solution: $8,954.24 
Quarterly 
12% ¸ 4 = 3%, 5 ´ 4 = 20 periods 
FV5 = $5,000 (1.806) 
FV5 = $9,030 
Calculator solution: $9,030.56 
(2) Annual Semiannual 
16%, 6 years 16% ¸ 2 = 8%, 6 ´ 2 = 12 periods 
FV6 = $5,000 (2.436) FV6 = $5,000 (2.518) 
FV6 = $12,180 FV6 = $12,590 
Calculator solution: $12,181.98 Calculator solution: $12,590.85
89 Gitman • Principles of Managerial Finance, Twelfth Edition 
Quarterly 
16% ¸ 4 = 4%, 6 ´ 4 = 24 periods 
FV6 = $5,000 (2.563) 
FV6 = $12,815 
Calculator solution: $12,816.52 
(3) Annual Semiannual 
20%, 10 years 20% ¸ 2 = 10%, 10 ´ 2 = 20 periods 
FV10 = $5,000 ´ (6.192) FV10 = $5,000 ´ (6.727) 
FV10 = $30,960 FV10 = $33,635 
Calculator solution: $30,958.68 Calculator solution: $33,637.50 
Quarterly 
20% ¸ 4 = 5%, 10 ´ 4 = 40 periods 
FV10 = $5,000 ´ (7.040) 
FV10 = $35,200 
Calculator solution: $35,199.94 
b. Effective interest rate: ieff = (1 + i/m)m – 1 
(1) Annual Semiannual 
ieff = (1 + 0.12/1)1 – 1 ieff = (1 + 12/2)2 – 1 
ieff = (1.12)1 – 1 ieff = (1.06)2 – 1 
ieff = (1.12) – 1 ieff = (1.124) – 1 
ieff = 0.12 = 12% ieff = 0.124 = 12.4% 
Quarterly 
ieff = (1 + 12/4)4 – 1 
ieff = (1.03)4 – 1 
ieff = (1.126) – 1 
ieff = 0.126 = 12.6% 
(2) Annual Semiannual 
ieff = (1 + 0.16/1)1 – 1 ieff = (1 + 0.16/2)2 – 1 
ieff = (1.16)1 – 1 ieff = (1.08)2 – 1 
ieff = (1.16) – 1 ieff = (1.166) – 1 
ieff = 0.16 = 16% ieff = 0.166 = 16.6% 
Quarterly 
ieff = (1 + 0.16/4)4 – 1 
ieff = (1.04)4 – 1 
ieff = (1.170) – 1 
ieff = 0.170 = 17%
Chapter 4 Time Value of Money 90 
(3) Annual Semiannual 
ieff = (1 + 0.20/1)1 – 1 ieff = (1 + 0.20/2)2 – 1 
ieff = (1.20)1 – 1 ieff = (1.10)2 – 1 
ieff = (1.20) – 1 ieff = (1.210) – 1 
ieff = 0.20 = 20% ieff = 0.210 = 21% 
Quarterly 
Ieff = (1 + 0.20/4)4 – 1 
Ieff = (1.05)4 – 1 
Ieff = (1.216) – 1 
Ieff = 0.216 = 21.6% 
P4-35. LG 5: Compounding frequency, time value, and effective annual rates 
Intermediate 
a. Compounding frequency: FVn = PV ´ FVIFi%,n 
A FV5 = $2,500 ´ (FVIF3%,10) B FV3 = $50,000 ´ (FVIF2%,18) 
FV5 = $2,500 ´ (1.344) FV3 = $50,000 ´ (1.428) 
FV5 = $3,360 FV3 = $71,400 
Calculator solution: $3,359.79 Calculator solution: $71,412.31 
C FV10 = $1,000 ´ (FVIF5%,10) D FV6 = $20,000 ´ (FVIF4%,24) 
FV10 = $1,000 ´ (1.629) FV6 = $20,000 ´ (2.563) 
FV10 = $16,290 FV6 = $51,260 
Calculator solution: $1,628.89 Calculator solution: $51,266.08 
b. Effective interest rate: ieff = (1 + i%/m)m – 1 
A ieff = (1 + 0.06/2)2 – 1 B ieff = (1 + 0.12/6)6 – 1 
ieff f = (1 + 0.03)2 – 1 ieff = (1 + 0.02)6 – 1 
ieff = (1.061) – 1 ieff = (1.126) – 1 
ieff = 0.061 = 06.1% ieff = 0.126 = 12.6% 
C ieff = (1 + 0.05/1)1 – 1 D ieff = (1 + 0.16/4)4 – 1 
ieff = (1 + 0.05)1 – 1 ieff = (1 + 0.04)4 – 1 
ieff = (1.05) – 1 ieff = (1.170) – 1 
ieff = 0.05 = 5% ieff = 0.17 = 17% 
c. The effective rates of interest rise relative to the stated nominal rate with increasing 
compounding frequency.
91 Gitman • Principles of Managerial Finance, Twelfth Edition 
P4-36. LG 5: Continuous compounding: FVcont. = PV ´ ex (e = 2.7183) 
Intermediate 
A FVcont. = $1,000 ´ e0.18 = $1,197.22 
B FVcont. = $ 600 ´ e1 = $1,630.97 
C FVcont. = $4,000 ´ e0.56 = $7,002.69 
D FVcont. = $2,500 ´ e0.48 = $4,040.19 
Note: If calculator doesn’t have ex key, use yx key, substituting 2.7183 for y. 
P4-37. LG 5: Personal finance: Compounding frequency and time value 
Challenge 
a. (1) FV10 = $2,000 ´ (FVIF8%,10) (2) FV10 = $2,000 ´ (FVIF4%,20) 
FV10 = $2,000 ´ (2.159) FV10 = $2,000 ´ (2.191) 
FV10 = $4,318 FV10 = $4,382 
Calculator solution: $4,317.85 Calculator solution: $4,382.25 
(3) FV10 = $2,000 ´ (FVIF0.022%,3650) (4) FV10 = $2,000 ´ (e0.8) 
FV10 = $2,000 ´ (2.232) FV10 = $2,000 ´ (2.226) 
FV10 = $4,464 FV10 = $4,452 
Calculator solution: $4,450.69 Calculator solution: $4,451.08 
b. (1) ieff = (1 + 0.08/1)1 – 1 (2) ieff = (1 + 0.08/2)2 – 1 
ieff = (1 + 0.08)1 – 1 ieff = (1 + 0.04)2 – 1 
ieff = (1.08) – 1 ieff = (1.082) – 1 
ieff = 0.08 = 8% ieff = 0.082 = 8.2% 
(3) ieff = (1 + 0.08/365)365 – 1 (4) ieff = (ek– 1) 
ieff = (1 + 0.00022)365 – 1 ieff = (e0.08– 1) 
ieff = (1.0833) – 1 ieff = (1.0833 – 1) 
ieff = 0.0833 = 8.33% ieff = 0.0833 = 8.33% 
c. Compounding continuously will result in $134 more dollars at the end of the 10 year period 
than compounding annually. 
d. The more frequent the compounding the larger the future value. This result is shown in part a 
by the fact that the future value becomes larger as the compounding period moves from 
annually to continuously. Since the future value is larger for a given fixed amount invested, 
the effective return also increases directly with the frequency of compounding. In part b we 
see this fact as the effective rate moved from 8% to 8.33% as compounding frequency moved 
from annually to continuously.
Chapter 4 Time Value of Money 92 
P4-38. LG 5: Personal finance: Comparing compounding periods 
Challenge 
a. FVn = PV ´ FVIFi%,n 
(1) Annually: FV = PV ´ FVIF12%,2 = $15,000 ´ (1.254) = $18,810 
Calculator solution: $18,816 
(2) Quarterly: FV = PV ´ FVIF3%,8 = $15,000 ´ (1.267) = $19,005 
Calculator solution: $19,001.55 
(3) Monthly: FV = PV ´ FVIF1%,24 = $15,000 ´ (1.270) = $19,050 
Calculator solution: $19,046.02 
(4) Continuously: FVcont. = PV ´ ex t 
FV = PV ´ 2.71830.24 = $15,000 ´ 1.27125 = $19,068.77 
Calculator solution: $19,068.74 
b. The future value of the deposit increases from $18,810 with annual compounding to $19,068.77 
with continuous compounding, demonstrating that future value increases as compounding 
frequency increases. 
c. The maximum future value for this deposit is $19,068.77, resulting from continuous 
compounding, which assumes compounding at every possible interval. 
P4-39. LG 3, 5: Personal finance: Annuities and compounding: FVAn = PMT ´ (FVIFAi%,n) 
Intermediate 
a. 
(1) Annual (2) Semiannual 
FVA10 = $300 ´ (FVIFA8%,10) FVA10 = $150 ´ (FVIFA4%,20) 
FVA10 = $300 ´ (14.487) FVA10 = $150 ´ (29.778) 
FVA10 = $4,346.10 FVA10 = $4,466.70 
Calculator solution: $4,345.97 Calculator solution: $4,466.71 
(3) Quarterly 
FVA10 = $75 ´.(FVIFA2%,40) 
FVA10 = $75 ´ (60.402) 
FVA10 = $4,530.15 
Calculator solution: $4,530.15 
b. The sooner a deposit is made the sooner the funds will be available to earn interest and 
contribute to compounding. Thus, the sooner the deposit and the more frequent the 
compounding, the larger the future sum will be.
93 Gitman • Principles of Managerial Finance, Twelfth Edition 
P4-40. LG 6: Deposits to accumulate growing future sum: = 
%, 
FVA 
PMT 
FVIFA 
n 
i n 
Basic 
Case Terms Calculation Payment 
A 12%, 3 yrs. PMT = $5,000 ¸ 3.374 = $1,481.92 
Calculator solution: $1,481.74 
B 7%, 20 yrs. PMT = $100,000 ¸ 40.995 = $2,439.32 
Calculator solution: $2,439.29 
C 10%, 8 yrs. PMT = $30,000 ¸ 11.436 = $2,623.29 
Calculator solution: $2,623.32 
D 8%, 12 yrs. PMT = $15,000 ¸ 18.977 = $ 790.43 
Calculator solution: $ 790.43 
P4-41. LG 6: Personal finance: Creating a retirement fund 
Intermediate 
a. PMT = FVA42 ¸ (FVIFA8%,42) b. FVA42 = PMT ´ (FVIFA8%,42) 
PMT = $220,000 ¸ (304.244) FVA42 = $600 ´ (304.244) 
PMT = $723.10 FVA42 = $182,546.40 
Calculator solution: $723.10 Calculator solution: $182,546.11 
P4-42. LG 6: Personal finance: Accumulating a growing future sum 
Intermediate 
FVn = PV ´ (FVIFi%,n) 
FV20 = $185,000 ´ (FVIF6%,20) 
FV20 = $185,000 ´ (3.207) 
FV20 = $593,295 = Future value of retirement home in 20 years. 
Calculator solution: $593,320.06 
PMT = FV ¸ (FVIFAi%,n) 
PMT = $593,295 ¸ (FVIFA10%,20) 
PMT = $593,295 ¸ (57.274) 
PMT = $10,358.89 
Calculator solution: $10,359.15 = annual payment required.
Chapter 4 Time Value of Money 94 
P4-43. LG 3, 6: Personal finance: Deposits to create a perpetuity 
Intermediate 
a. Present value of a perpetuity = PMT ´ (1 ¸ i) 
= $6,000 ´ (1 ¸ 0.10) 
= $6,000 ´ 10 
= $60,000 
b. PMT = FVA ¸ (FVIFA10%,10) 
PMT = $60,000 ¸ (15.937) 
PMT = $3,764.82 
Calculator solution: $3,764.72 
P4-44. LG 2, 3, 6: Personal finance: Inflation, time value, and annual deposits 
Challenge 
a. FVn = PV ´ (FVIFi%,n) 
FV20 = $200,000 ´ (FVIF5%,25) 
FV20 = $200,000 ´ (3.386) 
FV20 = $677,200 = Future value of retirement home in 25 years. 
Calculator solution: $677,270.99 
b. PMT = FV ¸ (FVIFAi%,n) 
PMT = $677,270.99 ¸ (FVIFA9%,25) 
PMT = $677,270.99 ¸ (84.699) 
PMT = $7,995.37 
Calculator solution: $7,996.03 = annual payment required. 
c. Since John will have an additional year on which to earn interest at the end of the 25 years his 
annuity deposit will be smaller each year. To determine the annuity amount John will first 
discount back the $677,200 one period. 
PV24 = $677,200´ 0.9174 = $621,263.28 
This is the amount John must accumulate over the 25 years. John can solve for his annuity 
amount using the same calculation as in part b. 
PMT = FV ¸ (FVIFAi%,n) 
PMT = $621,263.28 ¸ (FVIFA9%,25) 
PMT = $621,263.28 ¸ (84.699) 
PMT = $7,334.95 
Calculator solution: $7,335.81 = annual payment required. 
To check this value, multiply the annual payment by 1 plus the 9% discount rate. 
$7,335.81 (1.09) = $7996.03
95 Gitman • Principles of Managerial Finance, Twelfth Edition 
P4-45. LG 6: Loan payment: = 
%, 
PVA 
PMT 
PVIFAi n 
Basic 
Loan 
A PMT = $12,000 ¸ (PVIFA8%,3) B PMT = $60,000 ¸ (PVIFA12%,10) 
PMT = $12,000 ¸ 2.577 PMT = $60,000 ¸ 5.650 
PMT =  $4,656.58 PMT = $10,619.47 
Calculator solution: $4,656.40 Calculator solution: $10,619.05 
C PMT = $75,000 ¸ (PVIFA10%,30) D PMT = $4,000 ¸ (PVIFA15%,5) 
PMT = $75,000 ¸ 9.427 PMT = $4,000 ¸ 3.352 
PMT = $7,955.87 PMT = $1,193.32 
Calculator solution: $7,955.94 Calculator solution: $1,193.26 
P4-46. LG 6: Personal finance: Loan amortization schedule 
Intermediate 
a. PMT = $15,000 ¸ (PVIFA14%,3) 
PMT = $15,000 ¸ 2.322 
PMT = $6,459.95 
Calculator solution: $6,460.97 
b. 
End of 
Year 
Loan 
Payment 
Beginning of 
Year Principal 
Payments End of Year 
Interest Principal Principal 
1 $6,459.95 $15,000.00 $2,100.00 $4,359.95 $10,640.05 
2 6,459.95 10,640.05 1,489.61 4,970.34 5,669.71 
3 6,459.95 5,669.71 793.76 5,666.19 0 
(The difference in the last year’s beginning and ending principal is due to rounding.) 
c. Through annual end-of-the-year payments, the principal balance of the loan is declining, 
causing less interest to be accrued on the balance. 
P4-47. LG 6: Loan interest deductions 
Challenge 
a. PMT = $10,000 ¸ (PVIFA13%,3) 
PMT = $10,000 ¸ (2.361) 
PMT = $4,235.49 
Calculator solution: $4,235.22
Chapter 4 Time Value of Money 96 
b. 
End of 
Year 
Loan 
Payment 
Beginning of 
Year Principal 
Payments End of Year 
Interest Principal Principal 
1 $4,235.49 $10,000.00 $1,300.00 $2,935.49 $7,064.51 
2 4,235.49 7,064.51 918.39 3,317.10 3,747.41 
3 4,235.49 3,747.41 487.16 3,748.33 0 
(The difference in the last year’s beginning and ending principal is due to rounding.) 
P4-48. LG 6: Personal finance: Monthly loan payments 
Challenge 
a. PMT = $4,000 ¸ (PVIFA1%,24) 
PMT = $4,000 ¸ (21.243) 
PMT = $188.28 
Calculator solution: $188.29 
b. PMT = $4,000 ¸ (PVIFA0.75%,24) 
PMT = $4,000 ¸ (21.889) 
PMT = $182.74 
Calculator solution: $182.74 
P4-49. LG 6: Growth rates 
Basic 
a. PV = FVn ´ PVIFi%,n 
Case 
A PV = FV4 ´ PVIFk%,4yrs. B PV = FV9 ´ PVIFi%,9yrs. 
$500 = $800 ´ PVIFk%,4yrs $1,500 = $2,280 ´ PVIFk%,9yrs. 
0.625 = PVIFk%,4yrs 0.658 = PVIFk%,9yrs. 
12% < k < 13% 4% < k < 5% 
Calculator solution: 12.47% Calculator solution: 4.76% 
C PV = FV6 ´ PVIFi%,6 
$2,500 = $2,900 ´ PVIFk%,6 yrs. 
0.862 = PVIFk%,6yrs. 
2% < k < 3% 
Calculator solution: 2.50% 
b. 
Case 
A Same as in a 
B Same as in a 
C Same as in a 
c. The growth rate and the interest rate should be equal, since they represent the same thing.
97 Gitman • Principles of Managerial Finance, Twelfth Edition 
P4-50. LG 6: Personal finance: Rate of return: PVn = FVn ´ (PVIFi%,n) 
Intermediate 
a. PV = $2,000 ´ (PVIFi%,3yrs.) 
$1,500 = $2,000 ´ (PVIFi%,3 yrs.) 
0.75 = PVIFi%,3 yrs. 
10% < i < 11% 
Calculator solution: 10.06% 
b. Mr. Singh should accept the investment that will return $2,000 because it has a higher return 
for the same amount of risk. 
P4-51. LG 6: Personal finance: Rate of return and investment choice 
Intermediate 
a. A PV = $8,400 ´ (PVIFi%,6yrs.) B PV = $15,900 ´ (PVIFi%,15yrs.) 
$5,000 = $8,400 ´ (PVIFi%,6 yrs.) $5,000 = $15,900 ´ (PVIFi%,15yrs.) 
0.595 = PVIFi%,6 yrs. 0.314 = PVIFi%,15yrs. 
9% < i < 10% 8% < i < 9% 
Calculator solution: 9.03% Calculator solution: 8.02% 
C PV = $7,600 ´ (PVIFi%,4yrs.) D PV = $13,000 ´ (PVIFi%,10 yrs.) 
$5,000 = $7,600 ´ (PVIFi%,4 yrs.) $5,000 = $13,000 ´ (PVIFi%,10 yrs.) 
0.658 = PVIFi%,4 yrs. 0.385 = PVIFi%,10 yrs.. 
11% < i < 12% 10% < i < 11% 
Calculator solution: 11.04% Calculator solution: 10.03% 
b. Investment C provides the highest return of the four alternatives. Assuming equal risk for the 
alternatives, Clare should choose C. 
P4-52. LG 6: Rate of return-annuity: PVAn = PMT ´ (PVIFAi%,n) 
Basic 
$10,606 = $2,000 ´ (PVIFAi%,10 yrs.) 
5.303 = PVIFAi%,10 yrs. 
13% < i < 14% 
Calculator solution: 13.58% 
P4-53. LG 6: Personal finance: Choosing the best annuity: PVAn = PMT ´ (PVIFAi%,n) 
Intermediate 
a. Annuity A Annuity B 
$30,000 = $3,100 ´ (PVIFAi%,20 yrs.) $25,000 = $3,900 ´ (PVIFAi%,10 yrs.) 
9.677 = PVIFAi%,20 yrs. 6.410 = PVIFAi%,10 yrs. 
8% < i < 9% 9% < i < 10% 
Calculator solution: 8.19% Calculator solution: 9.03%
Chapter 4 Time Value of Money 98 
Annuity C Annuity D 
$40,000 = $4,200 ´ (PVIFAi%,15 yrs.) $35,000 = $4,000 ´ (PVIFAi%,12 yrs.) 
9.524 = PVFAi%,15 yrs. 8.75 = PVIFAi%,12 yrs. 
6% < i< 7% 5% < i < 6% 
Calculator solution: 6.3% Calculator solution: 5.23% 
b. Annuity B gives the highest rate of return at 9% and would be the one selected based upon 
Raina’s criteria. 
P4-54. LG 6: Personal finance: Interest rate for an annuity 
Challenge 
a. Defendants interest rate assumption 
$2,000,000 = $156,000 ´ (PVIFAi%,25 yrs.) 
12.821 = PVFAi%,25 yrs. 
5% < i < 6% 
Calculator solution: 5.97% 
b. Prosecution interest rate assumption 
$2,000,000 = $255,000 ´ (PVIFAi%,25 yrs.) 
7.843 = PVFAi%,25 yrs. 
i = 12% 
Calculator solution: 12.0% 
c. $2,000,000 = PMT ´ (PVIFA9%,25yrs.) 
$2,000,000 = PMT (9.823) 
PMT = $203,603.79 
Calculator solution: $203,612.50 
P4-55. LG 6: Personal finance: Loan rates of interest: PVAn = PMT ´ (PVIFAi%,n) 
Intermediate 
a. Loan A Loan B 
$5,000 = $1,352.81 ´ (PVIFAi%,5 yrs.) $5,000 = $1,543.21 ´ (PVIFAi%,4 yrs.) 
3.696 = PVIFAi%,5 yrs. 3.24 = PVIFAi%, 4 yrs. 
i = 11% i = 9% 
Loan C 
$5,000 = $2,010.45 ´ (PVIFAi%,3 yrs.) Calculator solutions are identical. 
2.487 = PVIFAk%,3 yrs. 
i = 10% 
b. Mr. Fleming should choose Loan B, which has the lowest interest rate.
99 Gitman • Principles of Managerial Finance, Twelfth Edition 
P4-56. LG 6: Number of years to equal future amount 
Intermediate 
A FV = PV ´ (FVIF7%,n yrs.) B FV = $12,000 ´ (FVIF5%,n yrs.) 
$1,000 = $300 ´ (FVIF7%,n yrs.) $15,000 = $12,000 ´ (FVIF5%,n yrs.) 
3.333 = FVIF7%,n yrs. 1.250 = FVIF5%,n yrs. 
17 < n < 18 4 < n < 5 
Calculator solution: 17.79 years Calculator solution: 4.573 years 
C FV = PV ´ (FVIF10%,n yrs.) D FV = $100 ´ (FVIF9%,n yrs.) 
$20,000 = $9,000 ´ (FVIF10%,n yrs.) $500 = $100 ´ (FVIF9%,n yrs.) 
2.222 = FVIF10%,n yrs. 5.00 = FVIF9%,n yrs. 
8 < n < 9 18 < n < 19 
Calculator solution: 8.38 years Calculator solution: 18.68 years 
E FV = PV ´ (FVIF15%,n yrs.) 
$30,000 = $7,500 ´ (FVIF15%,n yrs.) 
4.000 = FVIF15%,n yrs. 
9 < n < 10 
Calculator solution: 9.92 years 
P4-57. LG 6: Personal finance: Time to accumulate a given sum 
Intermediate 
a. 20,000 = $10,000 ´ (FVIF10%,n yrs.) b. 20,000 = $10,000 ´ (FVIF7%,n yrs.) 
2.000 = FVIF10%,n yrs. 2.000 = FVIF7%,n yrs. 
7 < n < 8 10 < n < 11 
Calculator solution: 7.27 years Calculator solution: 10.24 years 
c. 20,000 = $10,000 ´ (FVIF12%,n yrs.) 
2.000 = FVIF12%,n yrs. 
6 < n < 7 
Calculator solution: 6.12 years 
d. The higher the rate of interest the less time is required to accumulate a given future sum. 
P4-58. LG 6: Number of years to provide a given return 
Intermediate 
A PVA = PMT ´ (PVIFA11%,n yrs.) B PVA = PMT ´ (PVIFA15%,n yrs.) 
$1,000 = $250 ´ (PVIFA11%,n yrs.) $150,000 = $30,000 ´ (PVIFA15%,n yrs.) 
4.000 = PVIFA11%,n yrs. 5.000 = PVIFA15%,n yrs. 
5 < n < 6 9 < n < 10 
Calculator solution: 5.56 years Calculator solution: 9.92 years
Chapter 4 Time Value of Money 100 
C PVA = PMT ´ (PVIFA10%,n yrs.) D PVA = PMT ´ (PVIFA9%,n yrs.) 
$80,000 = $10,000 ´ (PVIFA10%,n yrs.) $600 = $275 ´ (PVIFA9%,n yrs.) 
8 = PVIFA10%,n yrs. 2.182 = PVIFA9%,n yrs. 
16 < n < 17 2 < n < 3 
Calculator solution: 16.89 years Calculator solution: 2.54 years 
E PVA = PMT ´ (PVIFA6%,n yrs.) 
$17,000 = $3,500 ´ (PVIFA6%,n yrs.) 
4.857 = PVIFA6%,n yrs. 
5 < n < 6 
Calculator solution: 5.91 years 
P4-59. LG 6: Personal finance: Time to repay installment loan 
Intermediate 
a. $14,000 = $2,450 ´ (PVIFA12%,n yrs.) 
5.714 = PVIFA12%,n yrs. 
10 < n < 11 
Calculator solution: 10.21 years 
b. $14,000 = $2,450 ´ (PVIFA9%,n yrs.) 
5.714 =  PVIFA9%,n yrs. 
8 < n < 9 
Calculator solution: 8.38 years 
c. $14,000 = $2,450 ´ (PVIFA15%,n yrs.) 
5.714 = PVIFA15%,n yrs. 
13 < n < 14 
Calculator solution: 13.92 years 
d. The higher the interest rate the greater the number of time periods needed to repay the loan 
fully. 
P4-60. Ethics problem 
Intermediate 
This is a tough issue. Even back in the Middle Ages, scholars debated the idea of a “just price.” 
The ethical debate hinges on (1) the basis for usury laws, (2) whether full disclosure is made of 
the true cost of the advance, and (3) whether customers understand the disclosures. Usury laws 
are premised on the notion that there is such a thing as an interest rate (price of credit) that is “too 
high.” A centuries-old fairness notion guides us into not taking advantage of someone in duress 
or facing an emergency situation. One must ask, too, why there are not market-supplied credit 
sources for borrowers, which would charge lower interest rates and receive an acceptable risk-adjusted 
return. On issues #2 and #3, there is no assurance that borrowers comprehend or are 
given adequate disclosures. See the box for the key ethics issues on which to refocus attention 
(some would view the objection cited as a smokescreen to take our attention off the true ethical 
issues in this credit offer).

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Managerial finance gitman_e12_ch04

  • 1.  Solutions to Problems P4-1. LG 1: Using a time line Basic a, b, and c d. Financial managers rely more on present value than future value because they typically make decisions before the start of a project, at time zero, as does the present value calculation. P4-2. LG 2: Future value calculation: FVn = PV ´ (1 + I)n Basic Case A FVIF12%,2 periods = (1 + 0.12)2 = 1.254 B FVIF6%,3 periods= (1 + 0.06)3 = 1.191 C FVIF9%,2 periods= (1 + 0.09)2 = 1.188 D FVIF3%,4 periods= (1 + 0.03)4 = 1.126 P4-3. LG 2: Future value tables: FVn = PV ´ (1 + I)n Basic Case A a. 2 = 1 ´ (1 + 0.07)n b. 4 = 1 ´ (1 + 0.07)n 2/1 = (1.07)n 4/1 = (1.07)n 2 = FVIF7%,n 4 = FVIF7%,n 10 years < n < 11 years 20 years < n < 21 years Nearest to 10 years Nearest to 20 years Case B
  • 2. 73 Gitman • Principles of Managerial Finance, Twelfth Edition a. 2 = 1 ´ (1 + 0.40)n b. 4 = (1 + 0.40)n 2 = FVIF40%,n 4 = FVIF40%,n 2 years < n < 3 years 4 years < n < 5 years Nearest to 2 years Nearest to 4 years Case C a. 2 = 1 ´ (1 + 0.20)n b. 4 = (1 + 0.20)n 2 = FVIF20%,n 4 = FVIF20%,n 3 years < n < 4 years 7 years < n < 8 years Nearest to 4 years Nearest to 8 years Case D a. 2 = 1 ´ (1 + 0.10)n b. 4 = (1 + 0.10)n 2 = FVIF10%,n 4 = FVIF40%,n 7 years < n < 8 years 14 years < n <15 years Nearest to 7 years Nearest to 15 years P4-4. LG 2: Future values: FVn = PV ´ (1 + I)n or FVn = PV ´ (FVIFi%,n) Intermediate Case Case A FV20 = PV ´ FVIF5%,20 yrs. B FV7 = PV ´ FVIF8%,7 yrs. FV20 = $200 ´ (2.653) FV7 = $4,500 ´ (1.714) FV20 = $530.60 FV7 = $7,713 Calculator solution: $530.66 Calculator solution: $7,712.21
  • 3. Chapter 4 Time Value of Money 74 C FV10 = PV ´ FVIF9%,10 yrs. D FV12 = PV ´ FVIF10%,12 yrs. FV10 = $10,000 ´ (2.367) FV12 = $25,000 ´ (3.138) FV10 = $23,670 FV12 = $78,450 Calculator solution: $23,673.64 Calculator solution: $78,460.71 E FV5 = PV ´ FVIF11%,5 yrs. F FV9 = PV ´ FVIF12%,9 yrs. FV5 = $37,000 ´ (1.685) FV9 = $40,000 ´ (2.773) FV5 = $62,345 FV9 = $110,920 Calculator solution: $62,347.15 Calculator solution: $110,923.15 P4-5. LG 2: Personal finance: Time value: FVn = PV ´ (1 + I)n or FVn = PV ´ (FVIFi%,n) Intermediate a. (1) FV3 = PV ´ (FVIF7%,3) b. (1) Interest earned = FV3 – PV FV3 = $1,500 ´ (1.225) Interest earned = $1,837.50 FV3 = $1,837.50 –$1,500.00 Calculator solution: $1,837.56 $337.50 (2) FV6 = PV ´ (FVIF7%,6) (2) Interest earned = FV6 – FV3 FV6 = $1,500 ´ (1.501) Interest earned = $2,251.50 FV6 = $2,251.50 –$1,837.50 Calculator solution: $2,251.10 $414.00 (3) FV9 = PV ´ (FVIF7%,9) (3) Interest earned = FV9 – FV6 FV9 = $1,500 ´ (1.838) Interest earned = $2,757.00 FV9 = $2,757.00 –$2,251.50 Calculator solution: $2,757.69 $505.50 c. The fact that the longer the investment period is, the larger the total amount of interest collected will be, is not unexpected and is due to the greater length of time that the principal sum of $1,500 is invested. The most significant point is that the incremental interest earned per 3-year period increases with each subsequent 3 year period. The total interest for the first 3 years is $337.50; however, for the second 3 years (from year 3 to 6) the additional interest earned is $414.00. For the third 3-year period, the incremental interest is $505.50. This increasing change in interest earned is due to compounding, the earning of interest on previous interest earned. The greater the previous interest earned, the greater the impact of compounding. P4-6. LG 2: Personal finance: Time value Challenge a. (1) FV5 = PV ´ (FVIF2%,5) (2) FV5 = PV ´ (FVIF4%,5) FV5 = $14,000 ´ (1.104) FV5 = $14,000 ´ (1.217)
  • 4. 75 Gitman • Principles of Managerial Finance, Twelfth Edition FV5 = $15,456.00 FV5 = $17,038.00 Calculator solution: $15,457.13 Calculator solution: $17,033.14 b. The car will cost $1,582 more with a 4% inflation rate than an inflation rate of 2%. This increase is 10.2% more ($1,582 ¸ $15,456) than would be paid with only a 2% rate of inflation. P4-7. LG 2: Personal finance: Time value Challenge Deposit Now: Deposit in 10 Years: FV40 = PV ´ FVIF9%,40 FV30 = PV10 ´ (FVIF9%,30) FV40 = $10,000 ´ (1.09)40 FV30 = PV10 ´ (1.09)30 FV40 = $10,000 ´ (31.409) FV30 = $10,000 ´ (13.268) FV40 = $314,090.00 FV30 = $132,680.00 Calculator solution: $314,094.20 Calculator solution: $132,676.78 You would be better off by $181,410 ($314,090 – $132,680) by investing the $10,000 now instead of waiting for 10 years to make the investment. P4-8. LG 2: Personal finance: Time value: FVn = PV ´ FVIFi%,n Challenge a. $15,000 = $10,200 ´ FVIFi%,5 b. $15,000 = $8,150 ´ FVIFi%,5 FVIFi%,5 = $15,000 ¸ $10,200 = 1.471 FVIFi%,5 = $15,000 ¸ $8,150 = 1.840 8% < i < 9% 12% < i < 13% Calculator solution: 8.02% Calculator solution: 12.98% c. $15,000 = $7,150 ´ FVIFi%,5 FVIFi%,5 = $15,000 ¸ $7,150 = 2.098 15% < i < 16% Calculator solution: 15.97% P4-9. LG 2: Personal finance: Single-payment loan repayment: FVn = PV ´ FVIFi%,n Intermediate a. FV1 = PV ´ (FVIF14%,1) b. FV4 = PV ´ (FVIF14%,4) FV1 = $200 ´ (1.14) FV4 = $200 ´ (1.689) FV1 = $228 FV4 = $337.80
  • 5. Chapter 4 Time Value of Money 76 Calculator solution: $228 Calculator solution: $337.79 c. FV8 = PV ´ (FVIF14%,8) FV8 = $200 ´ (2.853) FV8 = $570.60 Calculator solution: $570.52 P4-10. LG 2: Present value calculation: = 1 + PVIF (1 i)n Basic Case A PVIF = 1 ¸ (1 + 0.02)4 = 0.9238 B PVIF = 1 ¸ (1 + 0.10)2 = 0.8264 C PVIF = 1 ¸ (1 + 0.05)3 = 0.8638 D PVIF = 1 ¸ (1 + 0.13)2 = 0.7831 P4-11. LG 2: Present values: PV = FVn ´ (PVIFi%,n) Basic Case Calculator Solution A PV12%,4yrs = $7,000 ´ 0.636 = $4,452 $4,448.63 B PV8%, 20yrs = $28,000 ´ 0.215 = $6,020 $6,007.35 C PV14%,12yrs = $10,000 ´ 0.208 = $2,080 $2,075.59 D PV11%,6yrs = $150,000 ´ 0.535 = $80,250 $80,196.13 E PV20%,8yrs = $45,000 ´ 0.233 = $10,485 $10,465.56
  • 6. 77 Gitman • Principles of Managerial Finance, Twelfth Edition P4-12. LG 2: Present value concept: PVn = FVn ´ (PVIFi%,n) Intermediate a. PV = FV6 ´ (PVIF12%,6) b. PV = FV6 ´ (PVIF12%,6) PV = $6,000 ´ (.507) PV = $6,000 ´ (0.507) PV = $3,042.00 PV = $3,042.00 Calculator solution: $3,039.79 Calculator solution: $3,039.79 c. PV = FV6 ´ (PVIF12%,6) PV = $6,000 ´ (0.507) PV = $3,042.00 Calculator solution: $3,039.79 d. The answer to all three parts are the same. In each case the same questions is being asked but in a different way. P4-13. LG 2: Personal finance: Time value: PV = FVn ´ (PVIFi%,n) Basic Jim should be willing to pay no more than $408.00 for this future sum given that his opportunity cost is 7%. PV = $500 ´ (PVIF7%,3) PV = $500 ´ (0.816) PV = $408.00 Calculator solution: $408.15 P4-14. LG 2: Time value: PV = FVn ´ (PVIFi%,n) Intermediate PV = $100 ´ (PVIF8%,6) PV = $100 ´ (0.630) PV = $63.00 Calculator solution: $63.02
  • 7. Chapter 4 Time Value of Money 78 P4-15. LG 2: Personal finance: Time value and discount rates: PV = FVn ´ (PVIFi%,n) Intermediate a. (1) PV = $1,000,000 ´ (PVIF6%,10) (2) PV = $1,000,000 ´ (PVIF9%,10) PV = $1,000,000 ´ (0.558) PV = $1,000,000 ´ (0.422) PV = $558,000.00 PV = $422,000.00 Calculator solution: $558,394.78 Calculator solution: $422,410.81 (3) PV = $1,000,000 ´ (PVIF12%,10) PV = $1,000,000 ´ (0.322) PV = $322,000.00 Calculator solution: $321,973.24 b. (1) PV = $1,000,000 ´ (PVIF6%,15) (2) PV = $1,000,000 ´ (PVIF9%,15) PV = $1,000,000 ´ (0.417) PV = $1,000,000 ´ (0.275) PV = $417,000.00 PV = $275,000.00 Calculator solution: $417,265.06 Calculator solution: $274,538.04 (3) PV = $1,000,000 ´ (PVIF12%,15) PV = $1,000,000 ´ (0.183) PV = $183,000.00 Calculator solution: $182,696.26 c. As the discount rate increases, the present value becomes smaller. This decrease is due to the higher opportunity cost associated with the higher rate. Also, the longer the time until the lottery payment is collected, the less the present value due to the greater time over which the opportunity cost applies. In other words, the larger the discount rate and the longer the time until the money is received, the smaller will be the present value of a future payment. P4-16. Personal finance: LG 2: Time value comparisons of lump sums: PV = FVn ´ (PVIFi%,n) Intermediate a. A PV = $28,500 ´ (PVIF11%,3) B PV = $54,000 ´ (PVIF11%,9) PV = $28,500 ´ (0.731) PV = $54,000 ´ (0.391) PV = $20,833.50 PV = $21,114.00 Calculator solution: $20,838.95 Calculator solution: $21,109.94 C PV = $160,000 ´ (PVIF11%,20) PV = $160,000 ´ (0.124) PV = $19,840.00 Calculator solution: $19,845.43 b. Alternatives A and B are both worth greater than $20,000 in term of the present value. c. The best alternative is B because the present value of B is larger than either A or C and is also greater than the $20,000 offer.
  • 8. 79 Gitman • Principles of Managerial Finance, Twelfth Edition P4-17. LG 2: Personal finance: Cash flow investment decision: PV = FVn ´ (PVIFi%,n) Intermediate A PV = $30,000 ´ (PVIF10%,5) B PV = $3,000 ´ (PVIF10%,20) PV = $30,000 ´ (0.621) PV = $3,000 ´ (0.149) PV = $18,630.00 PV = $447.00 Calculator solution: $18,627.64 Calculator solution: $445.93 C PV = $10,000 ´ (PVIF10%,10) D PV = $15,000 ´ (PVIF10%,40) PV = $10,000 ´ (0.386) PV = $15,000 ´ (0.022) PV = $3,860.00 PV = $330.00 Calculator solution: $3,855.43 Calculator solution: $331.42 Purchase Do Not Purchase A B C D P4-18. LG 3: Future value of an annuity Intermediate a. Future value of an ordinary annuity vs. annuity due (1) Ordinary Annuity (2) Annuity Due FVAk%,n = PMT ´ (FVIFAk%,n) FVAdue = PMT ´ [(FVIFAk%,n ´ (1 + k)] A FVA8%,10 = $2,500 ´ 14.487 FVAdue = $2,500 ´ (14.487 ´ 1.08) FVA8%,10 = $36,217.50 FVAdue = $39,114.90 Calculator solution: $36,216.41 Calculator solution: $39,113.72 B FVA12%,6 = $500 ´ 8.115 FVAdue = $500 ´( 8.115 ´ 1.12) FVA12%,6 = $4,057.50 FVAdue = $4,544.40 Calculator solution: $4,057.59 Calculator solution: $4,544.51 C FVA20%,5 = $30,000 ´ 7.442 FVAdue = $30,000 ´ (7.442 ´ 1.20) FVA20%,5 = $223,260 FVAdue = $267,912 Calculator solution: $223,248 Calculator solution: $267,897.60 D FVA9%,8 = $11,500 ´ 11.028 FVAdue = $11,500 ´ (11.028 ´ 1.09) FVA9%,8 = $126,822 FVAdue = $138,235.98 Calculator solution: $126,827.45 Calculator solution: $138,241.92 E FVA14%,30 = $6,000 ´ 356.787 FVAdue = $6,000 ´ (356.787 ´ 1.14) FVA14%,30 = $2,140,722 FVAdue = $2,440,422.00 Calculator solution: $2,140,721.08 Calculator solution: $2,440,422.03 b. The annuity due results in a greater future value in each case. By depositing the payment at the beginning rather than at the end of the year, it has one additional year of compounding.
  • 9. Chapter 4 Time Value of Money 80 P4-19. LG 3: Present value of an annuity: PVn = PMT ´ (PVIFAi%,n) Intermediate a. Present value of an ordinary annuity vs. annuity due (1) Ordinary Annuity (2) Annuity Due PVAk%,n = PMT ´ (PVIFAi%,n) PVAdue = PMT ´ [(PVIFAi%,n ´ (1 + k)] A PVA7%,3 = $12,000 ´ 2.624 PVAdue = $12,000 ´ (2.624 ´ 1.07) PVA7%,3 = $31,488 PVAdue = $33,692 Calculator solution: $31,491.79 Calculator solution: $33,696.22 B PVA12%15 = $55,000 ´ 6.811 PVAdue = $55,000 ´ (6.811 ´ 1.12) PVA12%,15 = $374,605 PVAdue = $419,557.60 Calculator solution: $374,597.55 Calculator solution: $419,549.25 C PVA20%,9 = $700 ´ 4.031 PVAdue = $700 ´ (4.031 ´ 1.20) PVA20%,9 = $2,821.70 PVAdue = $3,386.04 Calculator solution: $2,821.68 Calculator solution: $3,386.01 D PVA5%,7 = $140,000 ´ 5.786 PVAdue = $140,000 ´ (5.786 ´ 1.05) PVA5%,7 = $810,040 PVAdue = $850,542 Calculator solution: $810,092.28 Calculator solution: $850,596.89 E PVA10%,5 = $22,500 ´ 3.791 PVAdue = $22,500 ´ (2.791 ´ 1.10) PVA10%,5 = $85,297.50 PVAdue = $93,827.25 Calculator solution: $85,292.70 Calculator solution: $93,821.97 b. The annuity due results in a greater present value in each case. By depositing the payment at the beginning rather than at the end of the year, it has one less year to discount back. P4-20. LG 3: Personal finance: Time value–annuities Challenge a. Annuity C (Ordinary) Annuity D (Due) FVAi%,n = PMT ´ (FVIFAi%,n) FVAdue = PMT ´ [FVIFAi%,n ´ (1 + i)] (1) FVA10%,10 = $2,500 ´ 15.937 FVAdue = $2,200 ´ (15.937 ´ 1.10) FVA10%,10 = $39,842.50 FVAdue = $38,567.54 Calculator solution: $39,843.56 Calculator solution: $38,568.57 (2) FVA20%,10 = $2,500 ´ 25.959 FVAdue = $2,200 ´ (25.959 ´ 1.20) FVA20%,10 = $64,897.50 FVAdue = $68,531.76 Calculator solution: $64,896.71 Calculator solution: $68,530.92
  • 10. 81 Gitman • Principles of Managerial Finance, Twelfth Edition b. (1) At the end of year 10, at a rate of 10%, Annuity C has a greater value ($39,842.50 vs. $38,567.54). (2) At the end of year 10, at a rate of 20%, Annuity D has a greater value ($68,531.76 vs. $64,897.50). c. Annuity C (Ordinary) Annuity D (Due) PVAi%,n = PMT ´ (FVIFAi%,n) PVAdue = PMT ´ [FVIFAi%,n ´ (1 + i)] (1) PVA10%,10 = $2,500 ´ 6.145 PVAdue = $2,200 ´ (6.145 ´ 1.10) PVA10%,10 = $15,362.50 PVAdue = $14,870.90 Calculator solution: $15,361.42 Calculator solution: $14,869.85 (2) PVA20%,10 = $2,500 ´ 4.192 PVAdue = $2,200 ´ (4.192 ´ 1.20) PVA20%,10 = $10,480 PVAdue = $11,066.88 Calculator solution: $10,481.18 Calculator solution: $11,068.13 d. (1) At the beginning of the 10 years, at a rate of 10%, Annuity C has a greater value ($15,362.50 vs. $14,870.90). (2) At the beginning of the 10 years, at a rate of 20%, Annuity D has a greater value ($11,066.88 vs. $10,480.00). e. Annuity C, with an annual payment of $2,500 made at the end of the year, has a higher present value at 10% than Annuity D with an annual payment of $2,200 made at the beginning of the year. When the rate is increased to 20%, the shorter period of time to discount at the higher rate results in a larger value for Annuity D, despite the lower payment. P4-21. LG 3: Personal finance: Retirement planning Challenge a. FVA40 = $2,000 ´ (FVIFA10%,40) b. FVA30 = $2,000 ´ (FVIFA10%,30) FVA40 = $2,000 ´ (442.593) FVA30 = $2,000 ´ (164.494) FVA40 = $885,186 FVA30 = $328,988 Calculator solution: $885,185.11 Calculator solution: $328,988.05 c. By delaying the deposits by 10 years the total opportunity cost is $556,198. This difference is due to both the lost deposits of $20,000 ($2,000 ´ 10yrs.) and the lost compounding of interest on all of the money for 10 years. d. Annuity Due: FVA40 = $2,000 ´ (FVIFA10%,40) ´ (1 + 0.10) FVA40 = $2,000 ´ (486.852) FVA40 = $973,704 Calculator solution: $973,703.62
  • 11. Chapter 4 Time Value of Money 82 FVA30 = $2,000 ´ (FVIFA10%,30) ´ (1.10) FVA30 = $2,000 ´ (180.943) FVA30 = $361,886 Calculator solution: $361,886.85 Both deposits increased due to the extra year of compounding from the beginning-of-year deposits instead of the end-of-year deposits. However, the incremental change in the 40 year annuity is much larger than the incremental compounding on the 30 year deposit ($88,518 versus $32,898) due to the larger sum on which the last year of compounding occurs. P4-22. LG 3: Personal finance: Value of a retirement annuity Intermediate PVA = PMT ´ (PVIFA9%,25) PVA = $12,000 ´ (9.823) PVA = $117,876.00 Calculator solution: $117,870.96 P4-23. LG 3: Personal finance: Funding your retirement Challenge a. PVA = PMT ´ (PVIFA11%,30) b. PV = FV ´ (PVIF9%,20) PVA = $20,000 ´ (8.694) PV = $173,880 ´ (0.178) PVA = $173,880.00 PV = $30,950.64 Calculator solution: $173,875.85 Calculator solution: $31,024.82 c. Both values would be lower. In other words, a smaller sum would be needed in 20 years for the annuity and a smaller amount would have to be put away today to accumulate the needed future sum. P4-24. LG 2, 3: Personal finance: Value of an annuity versus a single amount Intermediate a. PVAn = PMT ´ (PVIFAi%,n) PVA25 = $40,000 ´ (PVIFA5%,25) PVA25 = $40,000 ´ 14.094 PVA25 = $563,760 Calculator solution: $563,757.78 At 5%, taking the award as an annuity is better; the present value is $563,760, compared to receiving $500,000 as a lump sum.
  • 12. 83 Gitman • Principles of Managerial Finance, Twelfth Edition b. PVAn = $40,000 ´ (PVIFA7%,25) PVA25 = $40,000 ´ (11.654) PVA25 = $466,160 Calculator solution: $466,143.33 At 7%, taking the award as a lump sum is better; the present value of the annuity is only $466,160, compared to the $500,000 lump sum payment. c. Because the annuity is worth more than the lump sum at 5% and less at 7%, try 6%: PV25 = $40,000 ´ (PVIFA6%,25) PV25 = $40,000 ´ 12.783 PV25 = $511,320 The rate at which you would be indifferent is greater than 6%; about 6.25% Calculator solution: 6.24% P4-25. LG 3: Perpetuities: PVn = PMT ´ (PVIFAi%,¥) Basic a. b. Case PV Factor PMT ´ (PVIFAi%,¥) = PMT ´ (1 ¸ i) A 1 ¸ 0.08 = 12.50 $20,000 ´ 12.50 = $250,000 B 1 ¸ 0.10 = 10.00 $100,000 ´ 10.00 = $1,000,000 C 1 ¸ 0.06 = 16.67 $3,000 ´ 16.67 = $50,000 D 1 ¸ 0.05 = 20.00 $60,000 ´ 20.00 = $1,200,000 P4-26. LG 3: Personal finance: Creating an endowment Intermediate a. PV = PMT ´ (PVIFAi%,¥) b. PV = PMT ´ (PVIFAi%,¥) PV = ($600 ´ 3) ´ (1 ¸ i) PV = ($600 ´ 3) ´ (1 ¸ i) PV = $1,800 ´ (1 ¸ 0.06) PV = $1,800 ´ (1 ¸ 0.09) PV = $1,800 ´ (16.67) PV = $1,800 ´ (11.11) PV = $30,006 PV = $19,998 Calculator solution: $30,000 Calculator solution: $20,000
  • 13. Chapter 4 Time Value of Money 84 P4-27. LG 4: Value of a mixed stream Challenge a. Cash Flow Stream Year Number of Years to Compound FV = CF ´ FVIF12%,n Future Value A 1 3 $ 900 ´ 1.405 = $ 1,264.50 2 2 1,000 ´ 1.254 = 1,254.00 3 1 1,200 ´ 1.120 = 1,344 .00 $ 3,862 .50 Calculator solution: $ 3,862.84 B 1 5 $30,000 ´ 1.762 = $ 52,860.00 2 4 25,000 ´ 1.574 = 39,350.00 3 3 20,000 ´ 1.405 = 28,100.00 4 2 10,000 ´ 1.254 = 12,540.00 5 1 5,000 ´ 1.120 = 5,600 .00 $138,450 .00 Calculator solution: $138,450.79. C 1 4 $ 1,200 ´ 1.574 = $ 1,888.80 2 3 1,200 ´ 1.405 = 1,686.00 3 2 1,000 ´ 1.254 = 1,254.00 4 1 1,900 ´ 1.120 = 2,128 .00 $ 6,956 .80 Calculator solution: $ 6,956.54 b. If payments are made at the beginning of each period the present value of each of the end-of-period cash flow streams will be multiplied by (1 + i) to get the present value of the beginning-of- period cash flows. A $3,862.50 (1 + 0.12) = $4,326.00 B $138,450.00 (1 + 0.12) = $155,064.00 C $6,956.80 (1 + 0.12) = $7,791.62 P4-28. LG 4: Personal finance: Value of a single amount versus a mixed stream Intermediate Lump Sum Deposit FV5 = PV ´ (FVIF7%,5)) FV5 = $24,000 ´ (1.403) FV5 = $33,672.00 Calculator solution: $33,661.24
  • 14. 85 Gitman • Principles of Managerial Finance, Twelfth Edition Mixed Stream of Payments Beginning of Year Number of Years to Compound FV = CF ´ FVIF7%,n Future Value 1 5 $ 2,000 ´ 1.403 = $ 2,805.00 2 4 $ 4,000 ´ 1.311 = $ 5,243.00 3 3 $ 6,000 ´ 1.225 = $ 7,350.00 4 2 $ 8,000 ´ 1.145 = $ 9,159.00 5 1 $10,000 ´ 1.070 = $10,700 .00 $35,257 .00 Calculator solution: $35,257.75 Gina should select the stream of payments over the front-end lump sum payment. Her future wealth will be higher by $1,588. P4-29. LG 4: Value of mixed stream Basic Cash Flow Stream Year CF ´ PVIF12%,n = Present Value A 1 –$2000 ´ 0.893 = – $1,786 2 3,000 ´ 0.797 = 2,391 3 4,000 ´ 0.712 = 2,848 4 6,000 ´ 0.636 = 3,816 5 8,000 ´ 0.567 = 4,536 $11,805 Calculator solution: $11,805.51 B 1 $10,000 ´ 0.893 = $ 8,930 2–5 5,000 ´ 2.712a = 13,560 6 7,000 ´ 0.507 = 3,549 $26,039 Calculator solution: $26,034.58 *Sum of PV factors for years 2–5 C 1–5 $10,000 ´ 3.605b $36,050 6–10 8,000 ´ 2.045c 16,360 $52,410 Calculator solution: $52,411.34 a PVIFA for 12% over years 2 through 5 = (PVIFA 12% 5 years) – (PVIFA 12% 1 year) b PVIFA for 12% 5 years c (PVIFA for 12%,10 years) – (PVIFA for 12%,5 years)
  • 15. Chapter 4 Time Value of Money 86 P4-30. LG 4: PV-mixed stream Intermediate a. Cash Flow Stream Year CF ´ PVIF15%,n = Present Value A 1 $50,000 ´ 0.870 = $43,500 2 40,000 ´ 0.756 = 30,240 3 30,000 ´ 0.658 = 19,740 4 20,000 ´ 0.572 = 11,440 5 10,000 ´ 0.497 = 4,970 $109,890 Calculator solution: $109,856.33 B 1 $10,000 ´ 0.870 = $ 8,700 2 20,000 ´ 0.756 = 15,120 3 30,000 ´ 0.658 = 19,740 4 40,000 ´ 0.572 = 22,880 5 50,000 ´ 0.497 = 24,850 $ 91,290 Calculator solution: $ 91,272.98 b. Cash flow stream A, with a present value of $109,890, is higher than cash flow stream B’s present value of $91,290 because the larger cash inflows occur in A in the early years when their present value is greater, while the smaller cash flows are received further in the future. P4-31. LG 1, 4: Value of a mixed stream Intermediate a.
  • 16. 87 Gitman • Principles of Managerial Finance, Twelfth Edition b. Cash Flow Stream Year CF ´ PVIF12%,n = Present Value A 1 $30,000 ´ 0.893 = $ 26,790 2 25,000 ´ 0.797 = 19,925 3–9 15,000 ´ 3.639* = 54,585 10 10,000 ´ 0.322 = 3,220 $104,520 Calculator solution: $104,508.28 * The PVIF for this 7-year annuity is obtained by summing together the PVIFs of 12% for periods 3 through 9. This factor can also be calculated by taking the PVIFA12%,7 and multiplying by the PVIF12%,2. Alternatively, one could subtract PVIFA12%,2 from PVIFA12%,9. c. Harte should accept the series of payments offer. The present value of that mixed stream of payments is greater than the $100,000 immediate payment. P4-32. LG 5: Personal finance: Funding budget shortfalls Intermediate a. Year Budget Shortfall ´ PVIF8%,n = Present Value 1 $5,000 ´ 0.926 = $ 4,630 2 4,000 ´ 0.857 = 3,428 3 6,000 ´ 0.794 = 4,764 4 10,000 ´ 0.735 = 7,350 5 3,000 ´ 0.681 = 2,043 $22,215 Calculator solution: $22,214.03 A deposit of $22,215 would be needed to fund the shortfall for the pattern shown in the table. b. An increase in the earnings rate would reduce the amount calculated in part (a). The higher rate would lead to a larger interest being earned each year on the investment. The larger interest amounts will permit a decrease in the initial investment to obtain the same future value available for covering the shortfall.
  • 17. Chapter 4 Time Value of Money 88 P4-33. LG 4: Relationship between future value and present value-mixed stream Intermediate a. Present Value Year CF ´ PVIF5%,n = Present Value 1 $800 ´ 0.952 = $ 761.60 2 900 ´ 0.907 = 816.30 3 1,000 ´ 0.864 = 864.00 4 1,500 ´ 0.822 = 1,233.00 5 2,000 ´ 0.784 = 1,568.00 $5,242.90 Calculator solution: $5,243.17 b. The maximum you should pay is $5,242.90. c. A higher 7% discount rate will cause the present value of the cash flow stream to be lower than $5,242.90. P4-34. LG 5: Changing compounding frequency Intermediate a. Compounding frequency: FVn = PV ´ FVIFi%/m,n × m (1) Annual Semiannual 12%, 5 years 12% ¸ 2 = 6%, 5 ´ 2 = 10 periods FV5 = $5,000 ´ (1.762) FV5 = $5,000 ´ (1.791) FV5 = $8,810 FV5 = $8,955 Calculator solution: $8,811.71 Calculator solution: $8,954.24 Quarterly 12% ¸ 4 = 3%, 5 ´ 4 = 20 periods FV5 = $5,000 (1.806) FV5 = $9,030 Calculator solution: $9,030.56 (2) Annual Semiannual 16%, 6 years 16% ¸ 2 = 8%, 6 ´ 2 = 12 periods FV6 = $5,000 (2.436) FV6 = $5,000 (2.518) FV6 = $12,180 FV6 = $12,590 Calculator solution: $12,181.98 Calculator solution: $12,590.85
  • 18. 89 Gitman • Principles of Managerial Finance, Twelfth Edition Quarterly 16% ¸ 4 = 4%, 6 ´ 4 = 24 periods FV6 = $5,000 (2.563) FV6 = $12,815 Calculator solution: $12,816.52 (3) Annual Semiannual 20%, 10 years 20% ¸ 2 = 10%, 10 ´ 2 = 20 periods FV10 = $5,000 ´ (6.192) FV10 = $5,000 ´ (6.727) FV10 = $30,960 FV10 = $33,635 Calculator solution: $30,958.68 Calculator solution: $33,637.50 Quarterly 20% ¸ 4 = 5%, 10 ´ 4 = 40 periods FV10 = $5,000 ´ (7.040) FV10 = $35,200 Calculator solution: $35,199.94 b. Effective interest rate: ieff = (1 + i/m)m – 1 (1) Annual Semiannual ieff = (1 + 0.12/1)1 – 1 ieff = (1 + 12/2)2 – 1 ieff = (1.12)1 – 1 ieff = (1.06)2 – 1 ieff = (1.12) – 1 ieff = (1.124) – 1 ieff = 0.12 = 12% ieff = 0.124 = 12.4% Quarterly ieff = (1 + 12/4)4 – 1 ieff = (1.03)4 – 1 ieff = (1.126) – 1 ieff = 0.126 = 12.6% (2) Annual Semiannual ieff = (1 + 0.16/1)1 – 1 ieff = (1 + 0.16/2)2 – 1 ieff = (1.16)1 – 1 ieff = (1.08)2 – 1 ieff = (1.16) – 1 ieff = (1.166) – 1 ieff = 0.16 = 16% ieff = 0.166 = 16.6% Quarterly ieff = (1 + 0.16/4)4 – 1 ieff = (1.04)4 – 1 ieff = (1.170) – 1 ieff = 0.170 = 17%
  • 19. Chapter 4 Time Value of Money 90 (3) Annual Semiannual ieff = (1 + 0.20/1)1 – 1 ieff = (1 + 0.20/2)2 – 1 ieff = (1.20)1 – 1 ieff = (1.10)2 – 1 ieff = (1.20) – 1 ieff = (1.210) – 1 ieff = 0.20 = 20% ieff = 0.210 = 21% Quarterly Ieff = (1 + 0.20/4)4 – 1 Ieff = (1.05)4 – 1 Ieff = (1.216) – 1 Ieff = 0.216 = 21.6% P4-35. LG 5: Compounding frequency, time value, and effective annual rates Intermediate a. Compounding frequency: FVn = PV ´ FVIFi%,n A FV5 = $2,500 ´ (FVIF3%,10) B FV3 = $50,000 ´ (FVIF2%,18) FV5 = $2,500 ´ (1.344) FV3 = $50,000 ´ (1.428) FV5 = $3,360 FV3 = $71,400 Calculator solution: $3,359.79 Calculator solution: $71,412.31 C FV10 = $1,000 ´ (FVIF5%,10) D FV6 = $20,000 ´ (FVIF4%,24) FV10 = $1,000 ´ (1.629) FV6 = $20,000 ´ (2.563) FV10 = $16,290 FV6 = $51,260 Calculator solution: $1,628.89 Calculator solution: $51,266.08 b. Effective interest rate: ieff = (1 + i%/m)m – 1 A ieff = (1 + 0.06/2)2 – 1 B ieff = (1 + 0.12/6)6 – 1 ieff f = (1 + 0.03)2 – 1 ieff = (1 + 0.02)6 – 1 ieff = (1.061) – 1 ieff = (1.126) – 1 ieff = 0.061 = 06.1% ieff = 0.126 = 12.6% C ieff = (1 + 0.05/1)1 – 1 D ieff = (1 + 0.16/4)4 – 1 ieff = (1 + 0.05)1 – 1 ieff = (1 + 0.04)4 – 1 ieff = (1.05) – 1 ieff = (1.170) – 1 ieff = 0.05 = 5% ieff = 0.17 = 17% c. The effective rates of interest rise relative to the stated nominal rate with increasing compounding frequency.
  • 20. 91 Gitman • Principles of Managerial Finance, Twelfth Edition P4-36. LG 5: Continuous compounding: FVcont. = PV ´ ex (e = 2.7183) Intermediate A FVcont. = $1,000 ´ e0.18 = $1,197.22 B FVcont. = $ 600 ´ e1 = $1,630.97 C FVcont. = $4,000 ´ e0.56 = $7,002.69 D FVcont. = $2,500 ´ e0.48 = $4,040.19 Note: If calculator doesn’t have ex key, use yx key, substituting 2.7183 for y. P4-37. LG 5: Personal finance: Compounding frequency and time value Challenge a. (1) FV10 = $2,000 ´ (FVIF8%,10) (2) FV10 = $2,000 ´ (FVIF4%,20) FV10 = $2,000 ´ (2.159) FV10 = $2,000 ´ (2.191) FV10 = $4,318 FV10 = $4,382 Calculator solution: $4,317.85 Calculator solution: $4,382.25 (3) FV10 = $2,000 ´ (FVIF0.022%,3650) (4) FV10 = $2,000 ´ (e0.8) FV10 = $2,000 ´ (2.232) FV10 = $2,000 ´ (2.226) FV10 = $4,464 FV10 = $4,452 Calculator solution: $4,450.69 Calculator solution: $4,451.08 b. (1) ieff = (1 + 0.08/1)1 – 1 (2) ieff = (1 + 0.08/2)2 – 1 ieff = (1 + 0.08)1 – 1 ieff = (1 + 0.04)2 – 1 ieff = (1.08) – 1 ieff = (1.082) – 1 ieff = 0.08 = 8% ieff = 0.082 = 8.2% (3) ieff = (1 + 0.08/365)365 – 1 (4) ieff = (ek– 1) ieff = (1 + 0.00022)365 – 1 ieff = (e0.08– 1) ieff = (1.0833) – 1 ieff = (1.0833 – 1) ieff = 0.0833 = 8.33% ieff = 0.0833 = 8.33% c. Compounding continuously will result in $134 more dollars at the end of the 10 year period than compounding annually. d. The more frequent the compounding the larger the future value. This result is shown in part a by the fact that the future value becomes larger as the compounding period moves from annually to continuously. Since the future value is larger for a given fixed amount invested, the effective return also increases directly with the frequency of compounding. In part b we see this fact as the effective rate moved from 8% to 8.33% as compounding frequency moved from annually to continuously.
  • 21. Chapter 4 Time Value of Money 92 P4-38. LG 5: Personal finance: Comparing compounding periods Challenge a. FVn = PV ´ FVIFi%,n (1) Annually: FV = PV ´ FVIF12%,2 = $15,000 ´ (1.254) = $18,810 Calculator solution: $18,816 (2) Quarterly: FV = PV ´ FVIF3%,8 = $15,000 ´ (1.267) = $19,005 Calculator solution: $19,001.55 (3) Monthly: FV = PV ´ FVIF1%,24 = $15,000 ´ (1.270) = $19,050 Calculator solution: $19,046.02 (4) Continuously: FVcont. = PV ´ ex t FV = PV ´ 2.71830.24 = $15,000 ´ 1.27125 = $19,068.77 Calculator solution: $19,068.74 b. The future value of the deposit increases from $18,810 with annual compounding to $19,068.77 with continuous compounding, demonstrating that future value increases as compounding frequency increases. c. The maximum future value for this deposit is $19,068.77, resulting from continuous compounding, which assumes compounding at every possible interval. P4-39. LG 3, 5: Personal finance: Annuities and compounding: FVAn = PMT ´ (FVIFAi%,n) Intermediate a. (1) Annual (2) Semiannual FVA10 = $300 ´ (FVIFA8%,10) FVA10 = $150 ´ (FVIFA4%,20) FVA10 = $300 ´ (14.487) FVA10 = $150 ´ (29.778) FVA10 = $4,346.10 FVA10 = $4,466.70 Calculator solution: $4,345.97 Calculator solution: $4,466.71 (3) Quarterly FVA10 = $75 ´.(FVIFA2%,40) FVA10 = $75 ´ (60.402) FVA10 = $4,530.15 Calculator solution: $4,530.15 b. The sooner a deposit is made the sooner the funds will be available to earn interest and contribute to compounding. Thus, the sooner the deposit and the more frequent the compounding, the larger the future sum will be.
  • 22. 93 Gitman • Principles of Managerial Finance, Twelfth Edition P4-40. LG 6: Deposits to accumulate growing future sum: = %, FVA PMT FVIFA n i n Basic Case Terms Calculation Payment A 12%, 3 yrs. PMT = $5,000 ¸ 3.374 = $1,481.92 Calculator solution: $1,481.74 B 7%, 20 yrs. PMT = $100,000 ¸ 40.995 = $2,439.32 Calculator solution: $2,439.29 C 10%, 8 yrs. PMT = $30,000 ¸ 11.436 = $2,623.29 Calculator solution: $2,623.32 D 8%, 12 yrs. PMT = $15,000 ¸ 18.977 = $ 790.43 Calculator solution: $ 790.43 P4-41. LG 6: Personal finance: Creating a retirement fund Intermediate a. PMT = FVA42 ¸ (FVIFA8%,42) b. FVA42 = PMT ´ (FVIFA8%,42) PMT = $220,000 ¸ (304.244) FVA42 = $600 ´ (304.244) PMT = $723.10 FVA42 = $182,546.40 Calculator solution: $723.10 Calculator solution: $182,546.11 P4-42. LG 6: Personal finance: Accumulating a growing future sum Intermediate FVn = PV ´ (FVIFi%,n) FV20 = $185,000 ´ (FVIF6%,20) FV20 = $185,000 ´ (3.207) FV20 = $593,295 = Future value of retirement home in 20 years. Calculator solution: $593,320.06 PMT = FV ¸ (FVIFAi%,n) PMT = $593,295 ¸ (FVIFA10%,20) PMT = $593,295 ¸ (57.274) PMT = $10,358.89 Calculator solution: $10,359.15 = annual payment required.
  • 23. Chapter 4 Time Value of Money 94 P4-43. LG 3, 6: Personal finance: Deposits to create a perpetuity Intermediate a. Present value of a perpetuity = PMT ´ (1 ¸ i) = $6,000 ´ (1 ¸ 0.10) = $6,000 ´ 10 = $60,000 b. PMT = FVA ¸ (FVIFA10%,10) PMT = $60,000 ¸ (15.937) PMT = $3,764.82 Calculator solution: $3,764.72 P4-44. LG 2, 3, 6: Personal finance: Inflation, time value, and annual deposits Challenge a. FVn = PV ´ (FVIFi%,n) FV20 = $200,000 ´ (FVIF5%,25) FV20 = $200,000 ´ (3.386) FV20 = $677,200 = Future value of retirement home in 25 years. Calculator solution: $677,270.99 b. PMT = FV ¸ (FVIFAi%,n) PMT = $677,270.99 ¸ (FVIFA9%,25) PMT = $677,270.99 ¸ (84.699) PMT = $7,995.37 Calculator solution: $7,996.03 = annual payment required. c. Since John will have an additional year on which to earn interest at the end of the 25 years his annuity deposit will be smaller each year. To determine the annuity amount John will first discount back the $677,200 one period. PV24 = $677,200´ 0.9174 = $621,263.28 This is the amount John must accumulate over the 25 years. John can solve for his annuity amount using the same calculation as in part b. PMT = FV ¸ (FVIFAi%,n) PMT = $621,263.28 ¸ (FVIFA9%,25) PMT = $621,263.28 ¸ (84.699) PMT = $7,334.95 Calculator solution: $7,335.81 = annual payment required. To check this value, multiply the annual payment by 1 plus the 9% discount rate. $7,335.81 (1.09) = $7996.03
  • 24. 95 Gitman • Principles of Managerial Finance, Twelfth Edition P4-45. LG 6: Loan payment: = %, PVA PMT PVIFAi n Basic Loan A PMT = $12,000 ¸ (PVIFA8%,3) B PMT = $60,000 ¸ (PVIFA12%,10) PMT = $12,000 ¸ 2.577 PMT = $60,000 ¸ 5.650 PMT = $4,656.58 PMT = $10,619.47 Calculator solution: $4,656.40 Calculator solution: $10,619.05 C PMT = $75,000 ¸ (PVIFA10%,30) D PMT = $4,000 ¸ (PVIFA15%,5) PMT = $75,000 ¸ 9.427 PMT = $4,000 ¸ 3.352 PMT = $7,955.87 PMT = $1,193.32 Calculator solution: $7,955.94 Calculator solution: $1,193.26 P4-46. LG 6: Personal finance: Loan amortization schedule Intermediate a. PMT = $15,000 ¸ (PVIFA14%,3) PMT = $15,000 ¸ 2.322 PMT = $6,459.95 Calculator solution: $6,460.97 b. End of Year Loan Payment Beginning of Year Principal Payments End of Year Interest Principal Principal 1 $6,459.95 $15,000.00 $2,100.00 $4,359.95 $10,640.05 2 6,459.95 10,640.05 1,489.61 4,970.34 5,669.71 3 6,459.95 5,669.71 793.76 5,666.19 0 (The difference in the last year’s beginning and ending principal is due to rounding.) c. Through annual end-of-the-year payments, the principal balance of the loan is declining, causing less interest to be accrued on the balance. P4-47. LG 6: Loan interest deductions Challenge a. PMT = $10,000 ¸ (PVIFA13%,3) PMT = $10,000 ¸ (2.361) PMT = $4,235.49 Calculator solution: $4,235.22
  • 25. Chapter 4 Time Value of Money 96 b. End of Year Loan Payment Beginning of Year Principal Payments End of Year Interest Principal Principal 1 $4,235.49 $10,000.00 $1,300.00 $2,935.49 $7,064.51 2 4,235.49 7,064.51 918.39 3,317.10 3,747.41 3 4,235.49 3,747.41 487.16 3,748.33 0 (The difference in the last year’s beginning and ending principal is due to rounding.) P4-48. LG 6: Personal finance: Monthly loan payments Challenge a. PMT = $4,000 ¸ (PVIFA1%,24) PMT = $4,000 ¸ (21.243) PMT = $188.28 Calculator solution: $188.29 b. PMT = $4,000 ¸ (PVIFA0.75%,24) PMT = $4,000 ¸ (21.889) PMT = $182.74 Calculator solution: $182.74 P4-49. LG 6: Growth rates Basic a. PV = FVn ´ PVIFi%,n Case A PV = FV4 ´ PVIFk%,4yrs. B PV = FV9 ´ PVIFi%,9yrs. $500 = $800 ´ PVIFk%,4yrs $1,500 = $2,280 ´ PVIFk%,9yrs. 0.625 = PVIFk%,4yrs 0.658 = PVIFk%,9yrs. 12% < k < 13% 4% < k < 5% Calculator solution: 12.47% Calculator solution: 4.76% C PV = FV6 ´ PVIFi%,6 $2,500 = $2,900 ´ PVIFk%,6 yrs. 0.862 = PVIFk%,6yrs. 2% < k < 3% Calculator solution: 2.50% b. Case A Same as in a B Same as in a C Same as in a c. The growth rate and the interest rate should be equal, since they represent the same thing.
  • 26. 97 Gitman • Principles of Managerial Finance, Twelfth Edition P4-50. LG 6: Personal finance: Rate of return: PVn = FVn ´ (PVIFi%,n) Intermediate a. PV = $2,000 ´ (PVIFi%,3yrs.) $1,500 = $2,000 ´ (PVIFi%,3 yrs.) 0.75 = PVIFi%,3 yrs. 10% < i < 11% Calculator solution: 10.06% b. Mr. Singh should accept the investment that will return $2,000 because it has a higher return for the same amount of risk. P4-51. LG 6: Personal finance: Rate of return and investment choice Intermediate a. A PV = $8,400 ´ (PVIFi%,6yrs.) B PV = $15,900 ´ (PVIFi%,15yrs.) $5,000 = $8,400 ´ (PVIFi%,6 yrs.) $5,000 = $15,900 ´ (PVIFi%,15yrs.) 0.595 = PVIFi%,6 yrs. 0.314 = PVIFi%,15yrs. 9% < i < 10% 8% < i < 9% Calculator solution: 9.03% Calculator solution: 8.02% C PV = $7,600 ´ (PVIFi%,4yrs.) D PV = $13,000 ´ (PVIFi%,10 yrs.) $5,000 = $7,600 ´ (PVIFi%,4 yrs.) $5,000 = $13,000 ´ (PVIFi%,10 yrs.) 0.658 = PVIFi%,4 yrs. 0.385 = PVIFi%,10 yrs.. 11% < i < 12% 10% < i < 11% Calculator solution: 11.04% Calculator solution: 10.03% b. Investment C provides the highest return of the four alternatives. Assuming equal risk for the alternatives, Clare should choose C. P4-52. LG 6: Rate of return-annuity: PVAn = PMT ´ (PVIFAi%,n) Basic $10,606 = $2,000 ´ (PVIFAi%,10 yrs.) 5.303 = PVIFAi%,10 yrs. 13% < i < 14% Calculator solution: 13.58% P4-53. LG 6: Personal finance: Choosing the best annuity: PVAn = PMT ´ (PVIFAi%,n) Intermediate a. Annuity A Annuity B $30,000 = $3,100 ´ (PVIFAi%,20 yrs.) $25,000 = $3,900 ´ (PVIFAi%,10 yrs.) 9.677 = PVIFAi%,20 yrs. 6.410 = PVIFAi%,10 yrs. 8% < i < 9% 9% < i < 10% Calculator solution: 8.19% Calculator solution: 9.03%
  • 27. Chapter 4 Time Value of Money 98 Annuity C Annuity D $40,000 = $4,200 ´ (PVIFAi%,15 yrs.) $35,000 = $4,000 ´ (PVIFAi%,12 yrs.) 9.524 = PVFAi%,15 yrs. 8.75 = PVIFAi%,12 yrs. 6% < i< 7% 5% < i < 6% Calculator solution: 6.3% Calculator solution: 5.23% b. Annuity B gives the highest rate of return at 9% and would be the one selected based upon Raina’s criteria. P4-54. LG 6: Personal finance: Interest rate for an annuity Challenge a. Defendants interest rate assumption $2,000,000 = $156,000 ´ (PVIFAi%,25 yrs.) 12.821 = PVFAi%,25 yrs. 5% < i < 6% Calculator solution: 5.97% b. Prosecution interest rate assumption $2,000,000 = $255,000 ´ (PVIFAi%,25 yrs.) 7.843 = PVFAi%,25 yrs. i = 12% Calculator solution: 12.0% c. $2,000,000 = PMT ´ (PVIFA9%,25yrs.) $2,000,000 = PMT (9.823) PMT = $203,603.79 Calculator solution: $203,612.50 P4-55. LG 6: Personal finance: Loan rates of interest: PVAn = PMT ´ (PVIFAi%,n) Intermediate a. Loan A Loan B $5,000 = $1,352.81 ´ (PVIFAi%,5 yrs.) $5,000 = $1,543.21 ´ (PVIFAi%,4 yrs.) 3.696 = PVIFAi%,5 yrs. 3.24 = PVIFAi%, 4 yrs. i = 11% i = 9% Loan C $5,000 = $2,010.45 ´ (PVIFAi%,3 yrs.) Calculator solutions are identical. 2.487 = PVIFAk%,3 yrs. i = 10% b. Mr. Fleming should choose Loan B, which has the lowest interest rate.
  • 28. 99 Gitman • Principles of Managerial Finance, Twelfth Edition P4-56. LG 6: Number of years to equal future amount Intermediate A FV = PV ´ (FVIF7%,n yrs.) B FV = $12,000 ´ (FVIF5%,n yrs.) $1,000 = $300 ´ (FVIF7%,n yrs.) $15,000 = $12,000 ´ (FVIF5%,n yrs.) 3.333 = FVIF7%,n yrs. 1.250 = FVIF5%,n yrs. 17 < n < 18 4 < n < 5 Calculator solution: 17.79 years Calculator solution: 4.573 years C FV = PV ´ (FVIF10%,n yrs.) D FV = $100 ´ (FVIF9%,n yrs.) $20,000 = $9,000 ´ (FVIF10%,n yrs.) $500 = $100 ´ (FVIF9%,n yrs.) 2.222 = FVIF10%,n yrs. 5.00 = FVIF9%,n yrs. 8 < n < 9 18 < n < 19 Calculator solution: 8.38 years Calculator solution: 18.68 years E FV = PV ´ (FVIF15%,n yrs.) $30,000 = $7,500 ´ (FVIF15%,n yrs.) 4.000 = FVIF15%,n yrs. 9 < n < 10 Calculator solution: 9.92 years P4-57. LG 6: Personal finance: Time to accumulate a given sum Intermediate a. 20,000 = $10,000 ´ (FVIF10%,n yrs.) b. 20,000 = $10,000 ´ (FVIF7%,n yrs.) 2.000 = FVIF10%,n yrs. 2.000 = FVIF7%,n yrs. 7 < n < 8 10 < n < 11 Calculator solution: 7.27 years Calculator solution: 10.24 years c. 20,000 = $10,000 ´ (FVIF12%,n yrs.) 2.000 = FVIF12%,n yrs. 6 < n < 7 Calculator solution: 6.12 years d. The higher the rate of interest the less time is required to accumulate a given future sum. P4-58. LG 6: Number of years to provide a given return Intermediate A PVA = PMT ´ (PVIFA11%,n yrs.) B PVA = PMT ´ (PVIFA15%,n yrs.) $1,000 = $250 ´ (PVIFA11%,n yrs.) $150,000 = $30,000 ´ (PVIFA15%,n yrs.) 4.000 = PVIFA11%,n yrs. 5.000 = PVIFA15%,n yrs. 5 < n < 6 9 < n < 10 Calculator solution: 5.56 years Calculator solution: 9.92 years
  • 29. Chapter 4 Time Value of Money 100 C PVA = PMT ´ (PVIFA10%,n yrs.) D PVA = PMT ´ (PVIFA9%,n yrs.) $80,000 = $10,000 ´ (PVIFA10%,n yrs.) $600 = $275 ´ (PVIFA9%,n yrs.) 8 = PVIFA10%,n yrs. 2.182 = PVIFA9%,n yrs. 16 < n < 17 2 < n < 3 Calculator solution: 16.89 years Calculator solution: 2.54 years E PVA = PMT ´ (PVIFA6%,n yrs.) $17,000 = $3,500 ´ (PVIFA6%,n yrs.) 4.857 = PVIFA6%,n yrs. 5 < n < 6 Calculator solution: 5.91 years P4-59. LG 6: Personal finance: Time to repay installment loan Intermediate a. $14,000 = $2,450 ´ (PVIFA12%,n yrs.) 5.714 = PVIFA12%,n yrs. 10 < n < 11 Calculator solution: 10.21 years b. $14,000 = $2,450 ´ (PVIFA9%,n yrs.) 5.714 = PVIFA9%,n yrs. 8 < n < 9 Calculator solution: 8.38 years c. $14,000 = $2,450 ´ (PVIFA15%,n yrs.) 5.714 = PVIFA15%,n yrs. 13 < n < 14 Calculator solution: 13.92 years d. The higher the interest rate the greater the number of time periods needed to repay the loan fully. P4-60. Ethics problem Intermediate This is a tough issue. Even back in the Middle Ages, scholars debated the idea of a “just price.” The ethical debate hinges on (1) the basis for usury laws, (2) whether full disclosure is made of the true cost of the advance, and (3) whether customers understand the disclosures. Usury laws are premised on the notion that there is such a thing as an interest rate (price of credit) that is “too high.” A centuries-old fairness notion guides us into not taking advantage of someone in duress or facing an emergency situation. One must ask, too, why there are not market-supplied credit sources for borrowers, which would charge lower interest rates and receive an acceptable risk-adjusted return. On issues #2 and #3, there is no assurance that borrowers comprehend or are given adequate disclosures. See the box for the key ethics issues on which to refocus attention (some would view the objection cited as a smokescreen to take our attention off the true ethical issues in this credit offer).